symmetric willmore surfaces of revolution satisfying

Symmetric Willmore surfaces of revolution satisfying arbitrary

Dirichlet boundary data*

Anna Dall’Acqua∗, Steffen Frohlich, Hans-Christoph Grunau, Friedhelm Schieweck

August 9, 2010

Dedicated to Prof. E. Heinz on the occasion of his 85th birthday.

Abstract

We consider the Willmore boundary value problem for surfaces of revolution where, asDirichlet boundary conditions, any symmetric set of position and angle may be prescribed. Us-ing direct methods of the calculus of variations, we prove existence and regularity of minimisingsolutions. Moreover, we estimate the optimal Willmore energy and prove a number of qualita-tive properties of these solutions. Besides convexity-related properties we study in particularthe limit when the radii of the boundary circles converge to 0, while the “length” of the surfacesof revolution is kept fixed. This singular limit is shown to be the sphere, irrespective of theprescribed boundary angles.

These analytical investigations are complemented by presenting a numerical algorithm basedon C1-elements and numerical studies. They intensively interact with geometric constructionsin finding suitable minimising sequences for the Willmore functional.

Keywords. Dirichlet boundary conditions, Willmore surfaces of revolution.

AMS classification. 49Q10; 53C42, 35J65, 34L30.

Contents

1 Introduction 2

1.1 The Willmore problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Organisation of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Geometric background 8

2.1 Surfaces of revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Surfaces of revolution as elastic curves in the hyperbolic half plane . . . . . . . . . 8

2.3 Statement of the Willmore problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Existence result: The case β ≥ 0 10

3.1 The case αβ = 1: The circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 The case αβ > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.1 Monotonicity of the optimal energy . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.2 Properties of minimising sequences . . . . . . . . . . . . . . . . . . . . . . . 13

∗Financial support of “Deutsche Forschungsgemeinschaft” for the project “Randwertprobleme fur Willmoreflachen- Analysis, Numerik und Numerische Analysis” (DE 611/5.1) is gratefully acknowledged

1

3.2.3 Proof of the existence theorem . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 The case αβ < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15


3.3.2 Properties of minimising sequences . . . . . . . . . . . . . . . . . . . . . . . 16

3.3.3 Proof of the existence theorem . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Existence result: The case β < 0 18

4.1 The case α = αβ : The catenoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 The case α > αβ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2.1 First observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2.2 The case −β ≥ α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2.3 The case −β < α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 The case α < αβ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3.1 First observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31


4.3.3 The case α ≥ α∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3.4 The case α < α∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Convergence to the sphere for α ց 0 44

5.1 An upper bound for the energy as α ց 0 . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2 The limit of the energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3 The minimiser converges to the sphere . . . . . . . . . . . . . . . . . . . . . . . . . 48

6 Qualitative properties of minimisers and estimates of the energy 50

6.1 The case αβ > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.1.1 Bounds on the energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.1.2 On the sign of the hyperbolic curvature of minimisers . . . . . . . . . . . . 52

6.2 The case αβ < 1 and β ≥ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54



6.3 The case β < 0 and α ≥ αβ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55



6.4 The case β < 0 and α < αβ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56



7 Numerical studies and algorithms 57

1 Introduction

1.1 The Willmore problem

Given a smooth and immersed surface Γ ⊂ R3, the Willmore functional is defined by

W(Γ) :=

∫

Γ

H2 dA (1.1)

with H the mean curvature of the immersion and dA its area element.

2

The functional W is of geometric interest, and it models the elastic energy of thin shells orbiological membranes. It applies further in image processing and even in string theory (see e.g.[He, HP, Ni, KL, O]). In these applications one is usually concerned with minima or, moregenerally, with critical points of the Willmore functional. Such a critical point Γ ⊂ R

3 has tosatisfy the Willmore equation

∆ΓH + 2H(H2 − K) = 0 on Γ, (1.2)

where ∆Γ denotes the Laplace-Beltrami operator on Γ, and K is the Gauss curvature of the surface.A solution of this non-linear fourth-order differential equation is called Willmore surface.

Although introduced already in the 19th century (see e.g. [P]), it was Willmore’s work [Wi]which popularised again the investigation of the Willmore functional. Various existence andregularity results for closed Willmore surfaces of prescribed genus were extensively discussed inthe literature. We want to mention in particular Bauer-Kuwert and Simon [BK, Sn] for exis-tence of closed Willmore surfaces of prescribed genus, Kuwert-Schatzle and Leschke-Pedit-Pinkall[KS1, KS2, LPP] for constrained closed Willmore surfaces of fixed conformal class and Riviere [R]for a far reaching regularity result. We refer to [DDG] for a more extensive survey.

In the present paper we are interested in surfaces with boundaries. Therefore, we need toadd to (1.2) appropriate boundary conditions. A discussion of possible choices can be found inNitsche’s survey article [Ni]. In the present article we prescribe Dirichlet boundary conditions, i.e.∂Γ and the tangential spaces of Γ at ∂Γ. Nitsche’s work [Ni] contains also some existence resultsfor several kinds of boundary conditions. These are based on perturbation arguments and requiresevere smallness conditions on the boundary data, which are by no means explicit. Furthermore,using methods from geometric measure theory, Schatzle proved in [Sch] existence and regularity ofbranched Willmore immersions in S

n with prescribed Dirichlet boundary conditions. By workingin S

n, some compactness problems could be overcome. On the other hand, when pulling back theseimmersions to R

n it cannot be excluded that they contain the point ∞. Due to the generalityof his approach it seems to us that, in general, only little topological information of the solutioncan be extracted from the existence proof. However, under some explicit smallness condition onthe Willmore energy of suitable extensions of the Dirichlet boundary data, Schatzle’s solutionsare shown to be even connected and embedded. For numerical algorithms and numerical analysisfor boundary value problems for the Willmore equation and the corresponding parabolic flow wemention Deckelnick, Droske, Rumpf and Dziuk (see [DD, DR, Dz] and references therein).

To prove existence of a priori bounded solutions to boundary value problems for the Willmoreequation (1.2) with some specified further properties like e.g. the topological type or being a graphwithout imposing smallness conditions on the data seems to be a quite difficult task. Equation(1.2) is highly nonlinear and of fourth order and so, lacking any form of a general maximum orcomparison principle. Most of the well established techniques from second order problems likee.g. the De Giorgi-Nash-Moser theory seem to break down completely in higher order problems.In order to start working on a theory of classical bounded smooth solutions for the Willmoreboundary value problem we think that it is a good and appropriate strategy to investigate situ-ations enjoying symmetry. Although then, one has an underlying ordinary differential equation,understanding solvability of the corresponding boundary value problems is by no means straight-forward. In this spirit the one-dimensional Willmore problem or so called elastica were studied in[DG1, DG2]. Klaus Deckelnick and two of the authors investigated in [DDG] symmetric Willmoresurfaces of revolution where the position and zero slope were prescribed on the boundary. By anumber of refined geometric constructions it was possible to work with a priori bounded minimisingsequences. Although the differential equation is one-dimensional, the geometry is to a large extenttwo-dimensional: Great difficulties arising from the interaction between the principal curvaturesof the unknown surface are already present.

3

The previous work [DDG] was devoted to special Dirichlet boundary data. While the positionat the boundary could be prescribed arbitrarily, one had to restrict to a zero boundary angle.Arbitrary boundary angles are subject of the present paper.

1.2 Main results

In the present paper we will investigate a particular Dirichlet boundary value problem for (1.2).Namely, we consider surfaces of revolution Γ ⊂ R

3 which are generated by rotating a smoothfunction u : [−1, 1] → (0,∞) about the x = x1-axis. Then, Γ can be parametrised as follows:

(x, ϕ) 7→ f(x, ϕ) = (x, u(x) cos ϕ, u(x) sin ϕ), x ∈ [−1, 1], ϕ ∈ [0, 2π]. (1.3)

We consider the Willmore problem under symmetric Dirichlet boundary conditions where theheight u(±1) = α > 0 and an arbitrary angle u′(−1) = β = −u′(1), β ∈ R, are prescribed at theboundary. The case β = 0 has been studied in [DDG]. Our main result is the following.

Theorem 1.1 (Existence and regularity). For each α > 0 and each β ∈ R, there exists a positivesymmetric function u ∈ C∞([−1, 1], (0,∞)), i.e. u(x) > 0 and u(x) = u(−x), such that thecorresponding surface of revolution Γ ⊂ R

3 solves the Dirichlet problem for the Willmore equation

ΓH + 2H(H2 − K) = 0 in (−1, 1),

u(−1) = u(+1) = α, u′(−1) = −u′(+1) = β.(1.4)

The solution we find has the following additional properties:

1. If αβ > 1, then u′ < 0 in (0, 1] and |u′(x)| ≤ β for all x ∈ [−1, 1].

2. If αβ ≤ 1 and β ≥ 0, then u′ < 0 in (0, 1) and |u′(x)| ≤ 1α for all x ∈ [−1, 1].

3. If β < 0 and α arsinh(−β) ≥√

1 + β2, then u′ > 0 in (0, 1].

4. If β < 0 and α arsinh(−β) <√

1 + β2, then u has at most one critical point in (0, 1).

The proof is obtained by combining Theorems 3.11, 3.18, 4.17, 4.24, 4.39, 4.48 and Lemmas 3.1,3.20 and 4.1.

It may appear surprising that we find axially symmetric solutions of the Willmore boundaryvalue problem for all values of α > 0 and β ∈ R. For example, axially symmetric critical points ofthe area functional (i.e. minimal surfaces)

A(Γ) = 2π

1∫

−1

u(x)√

1 + u′(x)2 dx

exist only for u(1) = α ≥ α∗ where

α∗ :=1

b∗cosh(b∗) = 1.5088795 . . . (1.5)

and b∗ > 0 is the solution of the equation cosh(b∗) = b∗ sinh(b∗), b∗ = 1.1996786 . . .. Minimalsurfaces of revolution, so called catenoids, are obtained for any b ∈ (0,∞) by rotating the curvex 7→ 1

b cosh(bx) around the x-axis. Not only for boundary data α ∈ (0, α∗) these catenoids cease toexist, but according to [DHKW, Chapter 6.1, Theorem 3], there is no connected minimal surfacesolution at all – whether symmetric or not – for α < 1.

4

According to our result, for any set of symmetric Dirichlet boundary data, we always find atleast one solution to the Willmore boundary value problem. For non-symmetric Dirichlet data – e.g.u(1) 6= u(−1) – we expect a different picture. Analytical and numerical experiments suggest thatone may be forced to impose conditions on the data u(−1), u(1), u′(−1), u′(1) which deviate nottoo much from the symmetric setting. We feel that it might be even possible to prove nonexistencewithin the class of surfaces of revolutions generated by graphs for quite unsymmetric sets of data.For these data, however, existence may possibly still hold true in the class of parametric surfacesof revolution.

In order to prove our existence result Theorem 1.1, as in [DDG], we consider symmetric C1,1-functions satisfying the boundary conditions and we study the minimisation problem in this

class. In this setting, we prove that we may pass from arbitrary to suitable minimising sequencessatisfying strong a priori bounds. We obtain these bounds by explicit geometric constructionswhich lower the Willmore energy. A key observation in doing so is the correspondence between theWillmore functional on surfaces of revolution and a curvature functional (which we call hyperbolicWillmore functional) on curves in the hyperbolic half plane. The geometric constructions usegeodesics of the hyperbolic half plane as well as catenoids, i.e. minimal surfaces of revolution.The obtained a priori bounds on the elements of the suitably modified minimising sequence ensurethe required compactness and yield the desired existence result. In the setting of the hyperbolichalf plane a classification of possible curvature functions in terms of elliptic functions of the arclength of the unknown curves is available, see [LS1, LS2]. However, we did not see a possibility todevelop these results towards explicit formulae for boundary value problems (1.4). Moreover, wethink that the geometric constructions performed in the present paper help to a good extent tounderstand the geometric shape of minimisers.

It remains as an interesting question whether these solutions minimise the Willmore energyalso in the class of all immersed surfaces satisfying the same Dirichlet boundary conditions. Forβ 6= 0 and α → ∞ the energy bounds of Chapter 6 indicate that presumably this will not be thecase. We expect that there might be parametric Willmore surfaces of revolution with much smallerWillmore energy.

Uniqueness is a further issue we have to leave open. We think that similarly as in the one-dimensional analogous problem [DG1, Theorem 2] it should be possible to prove uniqueness andcontinuous dependence on data of energy minimising Willmore surfaces of revolution which aregenerated by graphs.

As can be seen from the statement of Theorem 1.1, the behaviour of those solutions of theWillmore equation constructed there depends not only on whether β ≥ 0 or β < 0. In both caseswe have to make further distinctions. It seems that we have to treat all these cases separately. Theswitch between the different cases occurs when having explicit solutions. These solutions markthe values of the parameters where the qualitative behaviour of solutions changes. If αβ = 1 thena solution is given by an arc of the circle with centre in the origin and going through the point(1, α). This is a geodesic in the hyperbolic half plane. The corresponding surface of revolutionis part of a sphere which is the simplest possible closed Willmore surface. These geodesics ofthe hyperbolic half plane play an important role when studying the case β ≥ 0. For β < 0 andα arsinh(−β) =

√1 + β2, the catenoid u(x) = cosh(bx)/b with b = arsinh(−β) is a minimal surface

solution. Catenoids come into play in our constructions in addition to the hyperbolic geodesicswhen studying the case β < 0. This interplay between two prototype Willmore surfaces gives riseto some technical difficulties. For β < 0 and |β| large, numerical calculations clearly display almostcatenoidal and almost spherical (hyperbolically geodesic) parts of solutions.

Conformal invariance is a key feature of the (hyperbolic) Willmore functional and of Willmoresurfaces. Rotation and translation are frequently employed, and scaling invariance is most impor-tant throughout the whole paper. On the other hand, inversions are not addressed here since in

5

most cases they do not preserve the particular shape (1.3) of surfaces of revolution generated bygraphs. Within this framework, only the relatively simple case αβ > 1 could have been reduced toresults in parts of the complementing cases, which are much more involved especially when β < 0.In particular, boundary data with β ≤ 1−α2

2α cannot be reduced to different cases because here,inversion does not yield graphs. But inversions are nevertheless quite interesting also here. De-pending on α, they may yield parametric Willmore surfaces of revolution which are not generatedby graphs because they approach the left boundary from the left and the right boundary from theright. This is remarkable in so far as the general discussion of parametric surfaces of revolution isexpected to be more difficult than that in the present paper.

Besides existence we also study further qualitative and asymptotic properties of solutions. Anatural question is what happens to the solutions constructed in Theorem 1.1 when β ∈ R is fixedand α goes to 0. We prove that they converge to the sphere centered at the origin with radius 1.

Theorem 1.2. Fix β ∈ R. For α > 0 let uα be a solution to problem (1.4) as constructed inTheorem 1.1. Then, uα converges for α ց 0 to x 7→

√1 − x2 in Cm

loc(−1, 1) for any m ∈ N.

For a proof see Theorem 5.8.With our method of proving existence of solutions we get also information on the qualitative

behaviour of the solutions. In particular, we can characterise the sign of the first derivative asstated in Theorem 1.1. Looking at the graph of a solution u : [−1, 1] → (0,∞) as a curve in thehyperbolic half plane, we study also the sign of its hyperbolic curvature. In Section 2.2 we recallsome basic facts from hyperbolic geometry. However, the meaning of the sign of the hyperboliccurvature κh[u](x) in (x, u(x)) is easily explained. One compares the graph of u in (x, u(x)) withthe tangential geodesic circle centered on the x-axis. Negative κh[u](x) means that the graph islocally inside this circle while κh[u](x) > 0 means that the graph of u is locally outside this circle.Concerning the sign of the hyperbolic curvature of our solutions we have the following result. Weskip the case αβ = 1, where the solution is a geodesic circle.

Theorem 1.3. For α > 0 and β ∈ R let u ∈ C∞([−1, 1], (0,∞)) be a solution to problem (1.4)as constructed in Theorem 1.1. Let κh[u] denote the hyperbolic curvature of the curve (x, u(x)) :x ∈ [−1, 1]. Then, κh[u] has the following sign properties:

1. If αβ > 1, then κh[u](0) < 0 and κh[u] has at most one change of sign in (0, 1).

2. If αβ < 1 and β ≥ 0, then κh[u] > 0 in (−1, 1).

3. If β < 0 and α arsinh(−β) >√

1 + β2, then κh[u](0) > 0 and κh[u] has at most one changeof sign in (0, 1).

4. If β < 0 and α arsinh(−β) ≤√

1 + β2, then κh[u] > 0 in (−1, 1).

The proof is obtained by combining Theorems 6.4, 6.7, 6.9 and 6.11.Numerical calculations give evidence to our feeling that in the case αβ > 1 the hyperbolic

curvature may indeed have a change of sign.It is not only in this respect that the analytical investigations of the present paper benefit

a lot from numerical simulations. Numerically calculated solutions help in finding qualitativeproperties of suitable minimising sequences while, at the same time, analytical insights help toidentify suitable initial data such that the numerical gradient flow method indeed converges. InChapter 7, we explain a C1-finite element method, which we think is natural in order to dealwith Dirichlet boundary conditions. It seems that so far, no C1-finite element algorithms areavailable for Willmore surfaces. Like in the analytic part we consider the present paper as a firststep also in numerical investigations of Dirichlet problems. We are confident that, basing upon

6

these experiences, we may develop C1-finite element algorithms also for graphs e.g. over generaltwo-dimensional domains. This will be subject of future research.

We remark that in particular the Navier boundary value problem is numerically well investi-gated, where the position of the surface and its mean curvature are prescribed at the boundary.See e.g. [DD, Dz] and references therein. In this case the Willmore boundary value problem maybe written as a second order system for the position and the mean curvature and continuous finiteelements may be used.

Droske and Rumpf [DR] proposed a level set formulation for the Dirichlet problem and forclosed Willmore surfaces and developed a corresponding piecewise linear continuous finite elementalgorithm.

1.3 Organisation of the paper

In Chapter 2 we recall some basic geometric notions which are relevant for our analysis, andformulate the minimisation problem for the Willmore functional as we shall study it. We explainthat the Willmore functional for surfaces of revolution Γ as in (1.3) corresponds to a functionaldefined on curves in the hyperbolic half plane. We call this second functional the “hyperbolicWillmore functional”. This observation was already made by Pinkall and Bryant-Griffiths andused in [Br, BG, LS2, DDG].

In Chapter 3 we prove Theorem 1.1 in the case β ≥ 0 taking advantage of the reformulationof the minimisation problem in the hyperbolic half plane. For αβ = 1 we have a part of a sphereas an explicit solution. We distinguish then the cases αβ > 1 and αβ < 1. In both cases wefirst prove monotonicity of the energy. The energy is increasing in α for αβ > 1, while it isdecreasing in α for αβ < 1. By geometric constructions we prove that we can restrict ourselves tominimising sequences satisfying strong a priori bounds, which are as in Theorem 1.1, Properties1 and 2 respectively. The key ingredient is to insert suitable parts of hyperbolic geodesic circles.The case αβ < 1 may be viewed as a direct generalisation of the result for β = 0 from [DDG]. Asfor estimates and existence we proceed exactly like there and are quite brief here for this reason.However, we improve it by showing that our solution even satisfies u′ < 0 in (0, 1). Obtaining apriori estimates in the case αβ > 1 is more involved since the geodesic circle through the boundarypoints does no longer serve as a comparison function.

In Chapter 4 we prove Theorem 1.1 in the case β < 0. For α = αβ :=√

1 + β2/arsinh(−β)a solution is the catenoid x 7→ cosh(bx)/b with b = arsinh(−β). Then, we distinguish the casesα > αβ and α < αβ . Here, the requisite geometric constructions in order to achieve strong enougha priori information on suitably modified minimising sequences do not only involve the hyperbolicgeodesics but also the catenoids as minimal surfaces of revolution. These constructions are differentnot only according to the cases α > αβ and α < αβ , but depend also on whether −β ≥ α or −β < αand whether α ≥ α∗ or α < α∗. The parameter α∗ = mincosh(b)/b : b ∈ (0,∞) refers to thesmallest boundary height where for some boundary angle one may have a catenoid as solution. If|β| becomes large and α small it turns out to be somehow delicate to prevent minimising sequencesfrom getting too close to 0 and to obtain bounds from below. Surprisingly, the case where α > αβ

and −β < α is special, because here we can prevent a possible loss of compactness only by furtherrestricting the class of admissible functions.

In Chapter 5 we study the behaviour of minimisers for α ց 0. We prove that our minimisersconverge locally uniformly in (−1, 1) to the sphere. In Chapter 6 we prove bounds on the Willmoreenergy and we study the sign of the hyperbolic curvature of the constructed solutions.

Chapter 7 gives a description of a C1-finite element algorithm for the underlying Willmoregradient flow. Moreover, numerical studies are performed, and we provide a series of picturesillustrating typical shapes of solutions within different parameter regimes.

7

2 Geometric background

2.1 Surfaces of revolution

We consider any function u ∈ C4([−1, 1], (0,∞)). Rotating the curve (x, u(x)) ⊂ R2 about the

x-axis generates a surface of revolution Γ ⊂ R3 which can be parametrised by

Γ : f(x, ϕ) =(x, u(x) cos ϕ, u(x) sinϕ

)∈ R

3 , x ∈ [−1, 1], ϕ ∈ [0, 2π).

The term “surface” always refers to the mapping f as well as to the set Γ. The condition u > 0implies that f is embedded in R

3 and in particular immersed.

Let κ1 and κ2 denote the principal curvatures of the surface Γ ⊂ R3, that is κ1 = −u′′(x)(1 +

u′(x)2)−3

2 and κ2 = (u(x)√

1 + u′(x)2)−1. Its mean curvature H and Gaussian curvature K are

H :=κ1 + κ2

2= − u′′(x)

2(1 + u′(x)2)3/2+

1

2u(x)√

1 + u′(x)2=

1

2u(x)u′(x)

(u(x)√

1 + u′(x)2

)′

,

K := κ1κ2 = − u′′(x)

u(1 + u′(x)2)2.

The Willmore energy of Γ defined in (1.1) is the integral over the surface of the mean curvaturesquared. In particular, written in terms of the function u it has the form

W(Γ) =π

2

1∫

−1

(u′′(x)

(1 + u′(x)2)3/2− 1

u(x)√

1 + u′(x)2

)2

u(x)√

1 + u′(x)2 dx. (2.1)

2.2 Surfaces of revolution as elastic curves in the hyperbolic half plane

Following [Br, BG], the construction of axially symmetric critical points Γ of the Willmore func-tional can be transformed to finding elastic curves in the upper half-plane R

2+ := (x, y) ∈ R

2 :y > 0 equipped with the hyperbolic metric ds2

h := 1y2 (dx2 + dy2). Geodesics are circular arcs

centered on the x-axis and lines parallel to the y-axis; the first will play a crucial role in this work.

Let s 7→ γ(s) = (γ1(s), γ2(s)), where we do not raise the indices, be a curve in R2+ parametrised

with respect to its arc length, i.e.

1 ≡ γ′1(s)

2 + γ′2(s)

2

γ2(s)2.

Then, its curvature is given by

κh(s) = −γ2(s)2

γ′2(s)

d

ds

(γ′

1(s)

γ2(s)2

)=

γ2(s)2

γ′1(s)

(1

γ2(s)+

d

ds

(γ′

2(s)

γ2(s)2

)). (2.2)

For graphs [−1, 1] ∋ x 7→ (x, u(x)) ∈ R2+, formula (2.2) yields

κh[u](x) = −u(x)2

u′(x)

d

dx

(1

u(x)√

1 + u′(x)2

)=

u(x)u′′(x)

(1 + u′(x)2)3/2+

1√1 + u′(x)2

. (2.3)

Using identity (2.3), we compute for the squared hyperbolic curvature times the hyperbolic line

8

element:

κh[u]2√

1 + u′2

u=

u′′

(1 + u′2)3/2+

1

u√

1 + u′2

2

u√

1 + u′2

=

u′′

(1 + u′2)3/2− 1

u√

1 + u′2

2

u√

1 + u′2 + 4u′′

(1 + u′2)3

2

= 4H2u√

1 + u′2 + 4u′′

(1 + u′2)3

2

.

We define the hyperbolic Willmore energy as the elastic energy of the graph of u in the hyperbolichalf plane and compare it with the original Willmore functional W(Γ) defined in (2.1).

Wh(u) :=

∫

γ

κh[u]2 dsh[u] :=

1∫

−1

κh[u]2√

1 + u′2

udx =

2

πW(Γ) + 4

1∫

−1

u′′

(1 + u′2)3/2dx, (2.4)

where Γ is the surface of revolution obtained by rotating the graph of u.

Lemma 2.1 (Duality of Wh(u) and W(Γ)). The hyperbolic energy Wh(u) of a curve u ∈ R2+ and

the Willmore energy of the corresponding surface of revolution Γ ⊂ R3 satisfy

Wh(u) =2

πW(Γ) + 4

[u′(x)√

1 + u′(x)2

]1

−1

.

This observation goes back to Pinkall (mentioned e.g. in [HJP]) and Bryant-Griffiths [Br, BG],see also [LS1, LS2]. The present derivation is adapted from [DDG, Section 2.2].

In proving Theorem 1.1, we benefit a lot from this duality between the Willmore functionaland the hyperbolic Willmore energy. We do not only take technical advantage from this result,but we think that switching between both functionals helps to a good extent in understandingunderlying geometric features.

Concerning the Euler-Lagrange equation for critical points of the hyperbolic Willmore func-tional Wh one has:

Lemma 2.2. Assume that u ∈ C4([−1, 1], (0,∞)) is such that for all ϕ ∈ C∞0 ([−1, 1], (0,∞)) one

has that 0 = ddtWh(u + tϕ)|t=0. Then, u satisfies the following Euler-Lagrange equation

u(x)√1 + u′(x)2

d

dx

(u(x)√

1 + u′(x)2κ′

h(x)

)− κh(x) +

1

2κh(x)3 = 0, x ∈ (−1, 1), (2.5)

with κh = κh[u] as defined in (2.3).

When parametrised by the hyperbolic arc length s, equation (2.5) takes the simple formd2

ds2 κh(s) − κh(s) + 12κh(s)3 = 0. This equation was discussed in detail in [LS1] and curvatures of

solutions were classified in terms of elliptic functions of the hyperbolic arc length s. However, wedo not see any possibility to solve directly and explicitly our Willmore boundary value problem(1.4) basing upon this classification.

2.3 Statement of the Willmore problem

The Willmore boundary value problem (1.4) will be solved by minimising the hyperbolic Willmorefunctional within the following class of functions:

9

Definition 2.3. For α > 0 and β ∈ R we introduce the function space

Nα,β :=u ∈ C1,1([−1, 1], (0,∞)) : u positive, symmetric, u(1) = α and u′(−1) = β

(2.6)

as well as

Mα,β := infWh(u) : u ∈ Nα,β

. (2.7)

Lemma 2.1 gives that

W(Γ) =π

2Wh(u) + 4π

β√1 + β2

for the surface Γ of revolution generated by u ∈ Nα,β . Since we are working with Dirichlet boundaryconditions we may switch between the two functionals depending on which one is more convenient.

In the following sections we will prove existence of solutions uα,β ∈ Nα,β ∩ C∞([−1, 1], R)such that Wh(uα,β) = Mα,β . Only in the case of parameters treated in Subsection 4.2.3, Nα,β

has for technical reasons to be replaced by a smaller set of admissible functions. The axiallysymmetric surface Γα,β which is generated by uα,β is solution of the Willmore boundary valueproblem (1.4). See [DG3, Lemma A.1] for an elementary calculation of the Euler-Lagrange equationin this particular setting. For a general survey on the Willmore functional, corresponding Euler-Lagrange equations and natural boundary conditions we refer to the survey article by Nitsche [Ni],cf. also [Th, p. 56]. The Euler-Lagrange equation for the Willmore functional in nonparametricform was already discussed by Poisson [P, p. 224].

Remark 2.4. The Willmore energy is invariant under rescaling. I.e. if u is a positive functionin C1,1([−r, r], (0,∞)) for some r > 0, then the function v ∈ C1,1([−1, 1], (0,∞)) defined byv(x) = u(rx)/r has the same hyperbolic Willmore energy as u, that is,

Wh(v) =

1∫

−1

κ2h[v] dsh[v] =

r∫

−r

κ2h[u] dsh[u].

Here, κh[u] is the hyperbolic curvature of u as defined in (2.3) and Wh(v) is the hyperbolic Willmoreenergy of v as defined in (2.4).

3 Existence result: The case β ≥ 0

In this section we consider β ≥ 0 and keep it fixed, while α varies in the positive real numbers.

For the value of α such that αβ = 1 we have an explicit solution of (1.4). This is the arc of thecircle with centre at the origin and going through the point (1, α). This solution is in particulara geodesic curve in the hyperbolic half plane. It marks the point where there is a change in thebehaviour of the energy. For αβ > 1 the energy Mα,β , defined in (2.7), is monotonically increasingin α, while for αβ < 1 it is monotonically decreasing in α.

Minimising sequences are suitably modified by means of parts of geodesic circles in order toachieve strong enough a priori estimates ensuring compactness. In this respect the case αβ > 1 ismore involved than the case αβ < 1, because here there is no canonical comparison function fromabove. However, one can pass to minimising sequences where the derivative is maximal in x = −1.The case αβ < 1 is quite similar to and contains the main result Theorem 1.1 from the previouswork [DDG] as a special case. However, this simplicity is due to referring to its main geometricconstruction. Here, a geodesic circle provides an obvious upper bound. Moreover, we prove anextra property of the solution u constructed there, namely that u′(x) < 0 on (0, 1).

10

3.1 The case αβ = 1: The circle

Here, we have an explicit solution.

Lemma 3.1. For each α > 0 and β such that αβ = 1, the part of the sphere Γ ⊂ R3 generated

as a surface of revolution by the function u(x) =√

1 + α2 − x2, x ∈ [−1, 1] solves the Dirichletproblem (1.4).

Moreover, the corresponding surface of revolution is the unique minimiser of the Willmorefunctional (1.1) among all axially symmetric surfaces generated by graphs of symmetric functionsin C1,1([−1, 1], (0,∞)) such that v(±1) = α and v′(1) = −β.

Proof. Since κh[u] ≡ 0 in [−1, 1], the claim follows from Lemma 2.1 and the definition of thehyperbolic Willmore functional in (2.4).

3.2 The case αβ > 1

3.2.1 Monotonicity of the optimal energy

In this paragraph we prove that the Willmore energy is increasing in α. The proof is divided intothe next four lemmas. First, we prove that it is enough to consider functions in Nα,β which aredecreasing in [0, 1]. The proof will refer to a main result of the previous work [DDG, Theorem3.8], which involves a number of refined geometric constructions. We emphasise that obviousconstructions like reflections do not yield the following result.

Lemma 3.2. For each u ∈ Nα,β with only finitely many critical points, we find a function v ∈ Nα,β

having at most as many critical points as u, with lower Willmore energy than u and satisfyingv′(x) ≤ 0 for all x ∈ [0, 1].

Proof. Assume that u does not have the claimed property. Then, there exists x0 ∈ (0, 1) such that[−x0, x0] is the largest possible symmetric interval with the property u′(x0) = 0 and u′(x) < 0 in(x0, 1]. Using a rescaled version of [DDG, Theorem 3.8] we substitute u|[−x0,x0] by a symmetricpositive C1,1-function defined on the same interval, having the same boundary values as u in x0,having lower Willmore energy than u|[−x0,x0], having at most as many critical points as u|[−x0,x0]

and decreasing in [0, x0]. The so obtained function v is element of Nα,β , it has at most as manycritical points as u, Wh(v) ≤ Wh(u) and v′(x) ≤ 0 in [0, 1].

In the proof we need only that β > 0. Notice further that one could substitute u|[−x0,x0] withan appropriately rescaled solution of the Willmore problem with β = 0 and height u(x0)/x0 asconstructed in [DDG, Theorem 1.1]. This statement, however, does not give control of the numberof critical points. With arguments introduced below we shall see – a posteriori – that we couldindeed achieve u′ < 0 on (0, x0).

In the next lemma, we construct for any u ∈ Nα,β which is decreasing in [0, 1] a function withthe same boundary values having lower Willmore energy than u and being defined in a largerinterval.

Lemma 3.3. Assume that u ∈ Nα,β has only finitely many critical points and satisfies u′(x) ≤ 0for all x ∈ [0, 1]. Then, for each ∈ [1, αβ), there exists a positive and symmetric functionu ∈ C1,1([−, ], (0,∞)) such that u() = α, u′

() = −β, u′(x) ≤ 0 for all x ∈ [0, ], u has at

most as many critical points as u, and, furthermore, one has

∫

−

κh[u]2 dsh[u] ≤ Wh(u).

11

Proof. The construction is similar to the one of [DDG, Lemma 3.3]. The situation there differsfrom the present one in the non-vanishing boundary conditions for u′. There we decrease the energyby shortening the interval, while here it is elongated.

Let r ∈ (0, 1) be a parameter. The (euclidian) normal to the graph of u in (r, u(r)) has direction(−u′(r), 1). The straight line generated by this normal intersects the x-axis left of r, since u isdecreasing. We take this intersection point (c(r), 0) as centre for a geodesic circular arc, wherethe radius is chosen such that this arc is tangential to the graph of u in (r, u(r)). In particular,the radius is given by the distance between (c(r), 0) and (r, u(r)). We build a new symmetricfunction with smaller hyperbolic curvature integral as follows: On [c(r), r] we take this geodesicarc, which has horizontal tangent in c(r), while on [r, 1] we take u. By construction, this functionis C1,1([c(r), 1], (0,∞)) and decreasing. We shift it such that c(r) is moved to 0, and extend this toan even function, which is again C1,1, now on a suitable interval [−ℓ(r), ℓ(r)], with ℓ(r) = 1− c(r)and c(r) = r + u(r)u′(r). This new function has the same boundary values as u, at most as manycritical points as u, and, by construction, a smaller curvature integral. Our construction yields theclaim since r 7→ ℓ(r) is continuous and such that limrց0 ℓ(r) = 1 and limrր1 ℓ(r) = αβ.

Remark 3.4. Notice that, by concavity of the geodesic circles, u′(x) ≥ −γ in [0, ] if u′(x) ≥ −γ

in [0, 1].

a)

α

arc ofgeodesiccircle

c(r) 0 r 1b)

α

0 1− −1

uu

Figure 1: Proof of Lemma 3.3.

By Lemma 3.2, we can remove the assumption that u′(x) ≤ 0 in [0, 1] from Lemma 3.3.

Lemma 3.5. Assume that u ∈ Nα,β has only finitely many critical points. Then, for each ∈[1, αβ), there exist a positive and symmetric function u ∈ C1,1([−, ], (0,∞)) such that u() = α,u′

() = −β, u′(x) ≤ 0 for all x ∈ [0, ], u has at most as many critical points as u, and,

furthermore, it holds that∫

−


Proof. By Lemma 3.2 there exists v in Nα,β having at most as many critical points as u, withlower Willmore energy than u and satisfying v′(x) ≤ 0 in [0, 1]. The claim follows from Lemma 3.3applied to v.

By rescaling we obtain:

Lemma 3.6. For each u ∈ Nα,β having only finitely many critical points and for each γ ∈ [β−1, α]there exists a symmetric function v ∈ C1,1([−1, 1], (0,∞)) having at most as many critical pointsas u and satisfying: v(±1) = γ, v′(1) = −β, v′(x) ≤ 0 in [0, 1] and Wh(v) ≤ Wh(u).

12

Proof. If γ ∈ (β−1, α] the claim follows from Lemma 3.5 by rescaling. If γ = β−1 we choosev(x) :=

√1 + γ2 − x2.

The previous lemma gives that the optimal Willmore energy Mα,β , defined in (2.7), is increasingin α.

Proposition 3.7. We have Meα,β ≥ Mα,β for all α, α such that α ≥ α ≥ 1β .

Proof. Since polynomials are dense in H2(−1, 1), a minimising sequence for Mα,β may be chosen inNα,β , which consists of symmetric and positive polynomials. Lemma 3.6 proves the statement.

In Proposition 6.2 we prove that even limα→∞ Mα,β = +∞.

3.2.2 Properties of minimising sequences

In the next two lemmas we introduce geometric constructions and show that on minimisingsequences, by possibly inserting parts of geodesic circles and rescaling, we may assume that0 ≥ u′(x) ≥ −β and x + u(x)u′(x) ≤ 0 for x ∈ [0, 1].

We first employ the elongation procedure of Lemma 3.3 and rescaling to achieve the derivativebounds.

Lemma 3.8. For each u ∈ Nα,β with only finitely many critical points there exists v ∈ Nα,β havingat most as many critical points as u, with lower Willmore energy than u and such that

−β ≤ v′(x) ≤ 0 for all x ∈ [0, 1].

Proof. By Lemma 3.2 there exists w ∈ Nα,β having at most as many critical points as u withlower Willmore energy than u and such that w′(x) ≤ 0 in [0, 1]. If moreover, w′(x) ≥ −β theclaim follows with v = w. Otherwise there exists a first x1 ∈ (0, 1) with w′(x1) = −β suchthat in particular w′(x) ≥ −β on [0, x1] . By using a scaled version of Lemma 3.3 we dilate thefunction w|[−x1,x1] by inserting an arc of a geodesic circle. For each ∈ (x1, w(x1)β) there existsw ∈ C1,1([−, ], (0,∞)) with lower Willmore energy than w|[−x1,x1], with at most as many criticalpoints as w|[−x1,x1] and such that w(±) = w(x1) and w′

() = −β. Notice that by concavity ofthe geodesic circles w′

(x) ≥ −β in [0, ]. We choose = w(x1)/α and v to be equal to w beingrescaled to the interval [−1, 1]. The choice of is such that we dilate the graph of w|[0,x1] until wereach the line y 7→ αy. This construction is illustrated in Figure 2.

We now add the property x + u(x)u′(x) ≤ 0 for x ∈ [0, 1] to those of the previous lemma bypossibly inserting a suitable part of a geodesic circle.

Lemma 3.9. For each u ∈ Nα,β having only finitely many critical points, there exists v ∈ Nα,β

having at most as many critical points as u, with lower Willmore energy than u and satisfying

0 ≥ v′(x) ≥ −β and 0 ≥ x + v(x)v′(x) for all x ∈ [0, 1].

Proof. By Lemma 3.8 there exists w ∈ Nα,β with lower Willmore energy than u, having at mostas many critical points as u and such that −β ≤ w′(x) ≤ 0 in [0, 1]. We consider the function ϕdefined in [0, 1] by

ϕ(x) := x + w(x)w′(x).

Note that ϕ(0) = 0 and ϕ(1) < 0. If ϕ ≤ 0 in [0, 1] then the claim follows with w = v. Otherwise,there exists x0 ∈ (0, 1) such that ϕ(x0) = 0 and ϕ 6≤ 0 in a left neighbourhood of x0. Then, thenormal line at (x0, w(x0)) to the graph of w passes through the origin and we can substitute w over

13

a)

replacemen

0 1x1

α

w

first point withw′(x) = −β

b) 0 1 x1

w

1. w, i.e.w|[0,x1] elongated

c) 0 1

w

1. w

2. v,by rescaling

v


[−x0, x0] by a geodesic circular arc lowering the hyperbolic Willmore energy. This new functionyields the claim. Notice that with this construction, due to the concavity of circles, the property−β ≤ w′ ≤ 0 is preserved and that we do not add critical points.

The following proposition summarises how by making use of Lemmas 3.8 and 3.9 we may passto minimising sequences satisfying suitable a priori bounds.

Proposition 3.10. Let (uk)k∈N be a minimising sequence for Mα,β in Nα,β such that each uk hasonly finitely many critical points. Then, there exists a minimising sequence (vk)k∈N ⊂ Nα,β suchthat for all k ∈ N it holds: vk has at most as many critical points as uk, Wh(vk) ≤ Wh(uk),

0 ≥ x + vk(x)v′k(x) and − β ≤ v′k(x) ≤ 0 for all x ∈ [0, 1]

and√

1 + α2 − x2 ≤ vk(x) ≤√

(α + β)2 − x2 for all x ∈ [−1, 1].(3.1)

3.2.3 Proof of the existence theorem

The proof of the following theorem follows the lines of the proof of [DDG, Theorem 3.9].

Theorem 3.11 (Existence and regularity). For each α > 0 and β such that αβ > 1 there existsa function u ∈ C∞([−1, 1], (0,∞)) such that the corresponding surface of revolution Γ ⊂ R

3 solvesthe Dirichlet problem (1.4). This solution is positive and symmetric, and it has the followingproperties:

−β ≤ u′(x) < 0 and x + u(x)u′(x) < 0 in (0, 1]

as well as√

1 + α2 − x2 ≤ u(x) ≤√

(α + β)2 − x2 in [−1, 1].(3.2)

Proof. Let (uk)k∈N ∈ Nα,β be a minimising sequence for Mα,β such that Wh(uk) ≤ Mα,β + 1 forall k ∈ N. By the density of polynomials in H2(−1, 1) and Proposition 3.10 we may assume thateach element uk of the minimising sequence satisfies (3.1). We can estimate the Willmore energy

14

from below as follows:

Wh(uk) =

1∫

−1

u′′k(x)2uk(x)

(1 + u′k(x)2)

5

2

dx + 2

1∫

−1

u′′k(x)

(1 + u′k(x)2)

3

2

dx +

1∫

−1

1

uk(x)√

1 + u′k(x)2

dx

≥ α

(1 + β2)5

2

1∫

−1

u′′k(x)2 dx − 4β√

1 + β2.

Thus, (uk)k∈N is uniformly bounded in H2(−1, 1), and, eventually, after passing to a subsequence,Rellich’s embedding theorem ensures the existence of u ∈ H2(−1, 1) such that

uk u in H2(−1, 1) and uk → u ∈ C1([−1, 1], (0,∞)).

Making use of the strong convergence in C1([−1, 1]) and the weak convergence in H2(−1, 1) of thesequence (uk)k∈N, we have

Mα,β + o(1) = Wh(uk) =

1∫

−1

u′′2k u

(1 + u′2)5

2

dx +

1∫

−1

1

u√

1 + u′2 dx − 4β√1 + β2

+ o(1)

≥1∫

−1

u′′2u

(1 + u′2)5

2

dx +

1∫

−1

1

u√

1 + u′2 dx − 4β√1 + β2

+ o(1) = Wh(u) + o(1).

Thus, u minimises Wh in the class of all positive and symmetric H2(−1, 1)-functions v satisfyingv(±1) = α, v′(+1) = −β, and, therefore, u weakly solves (2.5). Moreover, since the elements ofthe minimising sequence satisfy (3.1) then u satisfies x + u(x)u′(x) ≤ 0 and −β ≤ u′(x) ≤ 0 in(0, 1]. From the first inequality it follows that u′ < 0 in (0, 1].

The proof of smoothness of the solution is exactly as in [DDG, Theorem 3.9, Step 2].Finally we show that u satisfies x + u(x)u′(x) < 0 in (0, 1]. Indeed, if x0 + u(x0)u

′(x0) = 0for some x0 ∈ (0, 1] then reasoning as in Lemma 3.9 and using that Mα,β = Wh(u), we see that uequals an arc of a geodesic circle in [−x0, x0]. But u being a solution of (2.5) implies by uniquenessof the initial value problem that u is a geodesic circular arc on [−1, 1]. But such an arc cannotsatisfy the boundary conditions when αβ > 1.

In Lemma 6.3 we prove further that u′ is a decreasing function in [0, 1].

Proposition 3.12. Let αβ > 1. Then, Meα,β > Mα,β for all α such that α > α.

Proof. Let ueα be a solution of (1.4) for boundary values α and β as constructed in Theorem 3.11.By proceeding as in Lemma 3.5, i.e. inserting an appropriately chosen circular arc, we get afunction v ∈ Nα,β such that Wh(v) ≤ Wh(ueα). We prove that this inequality is in fact strict. Aswe have seen in the proof of Theorem 3.11, ueα cannot be equal to an arc of a geodesic circle inan interval. Hence, by introducing a piece of a geodesic circle the energy strictly decreases. Theclaim follows since Wh(v) ≥ Mα,β . Notice that, for the same reason, also this last inequality isstrict.

3.3 The case αβ < 1

The method of proof is related to that for the case αβ > 1 but much simpler. The results are,in some sense, dual. For the monotonicity of the energy in the case αβ > 1, we have constructed

15

a function with lower Willmore energy and defined in a bigger interval. Now, with the sameconstruction, the function is defined in a shorter interval. Moreover, we show that in this case wecan confine ourselves to functions satisfying x + u(x)u′(x) ≥ 0 in (0, 1]. A lower bound for thederivative follows directly from this inequality. We proceed quite similarly as in the case β = 0,which was discussed in the previous paper [DDG] and which is included here.


In this case the Willmore energy is decreasing in α. The proof is as in paragraph 3.2.1. For thesake of conciseness we formulate only the results.

Lemma 3.13. Assume that u ∈ Nα,β has only finitely many critical points. Then, for each ∈ [αβ, 1], there exist a positive and symmetric function u ∈ C1,1([−, ], (0,∞)) such thatu() = α, u′

() = −β, u′(x) ≤ 0 for all x ∈ [0, ], u has at most as many critical points as u,

and, furthermore, one has∫

−


In the next two results, for β = 0 we interpret 1/β as ∞.

Lemma 3.14. For each u ∈ Nα,β having only finitely many critical points and for each γ ∈ [α, β−1]there exists a symmetric function v ∈ C1,1([−1, 1], (0,∞)) having at most as many critical pointsas u and satisfying: v(±1) = γ, v′(1) = −β, v′(x) ≤ 0 in [0, 1] and Wh(v) ≤ Wh(u).

Proposition 3.15. It holds that Meα,β ≥ Mα,β for all α, α such that 0 < α ≤ α ≤ 1β .

3.3.2 Properties of minimising sequences

In the next lemma we show that we can restrict ourselves to functions which are decreasing in (0, 1]and satisfy x + u(x)u′(x) ≥ 0 in (0, 1]. A priori bounds follow directly from these observations.

Lemma 3.16. For each u ∈ Nα,β with only finitely many critical points there exists v ∈ Nα,β withlower Willmore energy than u, having at most as many critical points as u and such that

0 ≤ x + v(x)v′(x) and v′(x) ≤ 0 for all x ∈ [0, 1].

Proof. By Lemma 3.2 and the following remark there exists w ∈ Nα,β with lower Willmore energythan u, having at most as many critical points as u and such that w′(x) ≤ 0 in [0, 1]. Let usconsider the function ϕ defined in [0, 1] by ϕ(x) := x + w(x)w′(x). Note that ϕ(0) = 0 andϕ(1) > 0. If ϕ ≥ 0 in [0, 1] then the claim follows with w = v. Otherwise, there exists x0 ∈ (0, 1)such that ϕ(x0) = 0 and ϕ ≥ 0 in (x0, 1]. Then, the normal line at (x0, w(x0)) to the graphof w passes through the origin and we can substitute w over [−x0, x0] by a geodesic circular arclowering the hyperbolic Willmore energy. The new function so obtained yields the claim. Withthis construction we do not add critical points.

The following proposition characterises suitably modified minimising sequences.

Proposition 3.17. Let (uk)k∈N be a minimising sequence for Mα,β in Nα,β such that each uk hasonly finitely many critical points. Then, there exists a minimising sequence (vk)k∈N ⊂ Nα,β suchthat for all k ∈ N: vk has at most as many critical points as uk, Wh(vk) ≤ Wh(uk) and satisfying:

0 ≤ x + vk(x)v′k(x), v′k(x) ≤ 0 and v′k(x) ≥ − xvk(x) ≥ − x

α for all x ∈ [0, 1],

and α ≤ vk(x) ≤√

1 + α2 − x2 for all x ∈ [−1, 1].(3.3)

16

3.3.3 Proof of the existence theorem

Thanks to Proposition 3.17 we prove now existence of a solution. The following result is a directgeneralisation of [DDG, Theorem 1.1]. Its proof appears to be relatively simple but one shouldobserve that via Lemma 3.2 the main constructions of [DDG, Theorem 3.8] are essentially used.

Theorem 3.18 (Existence and regularity). For each α > 0 and each β ≥ 0 such that αβ < 1there exists a function u ∈ C∞([−1, 1], (0,∞)) such that the corresponding surface of revolutionΓ ⊂ R

3 solves the Dirichlet problem (1.4). This solution is positive and symmetric, and it has thefollowing properties:

−x

α≤ u′(x) ≤ 0 and x + u(x)u′(x) > 0 in (0, 1], and α ≤ u(x) ≤

√1 + α2 − x2 in [−1, 1]. (3.4)

Proof. Let (uk)k∈N ∈ Nα,β be a minimising sequence for Mα,β such that Wh(uk) ≤ Mα,β + 1 forall k ∈ N. By Proposition 3.17 and the density of polynomials in H2(−1, 1) we may assume thateach element uk of the minimising sequence satisfies (3.3). The rest of the proof is on the sameline as that of Theorem 3.11.

Proceeding as in the proof of Proposition 3.12, one can show that the energy is strictly de-creasing.

Proposition 3.19. Let α > 0, β ≥ 0 and αβ < 1. Then, Meα,β > Mα,β for all α ∈ (0, α).

Also in this case, we can prove an additional qualitative information on our solution of (1.4)constructed in Theorem 3.18, namely that u′ < 0 in (0, 1). This property is expected but here, itis slightly more involved to prove it when compared with the dual case αβ > 1. It will prove tobe helpful also for the constructions in the case β < 0.

Lemma 3.20. Let u be a solution of (1.4) minimising the hyperbolic Willmore energy in Nα,β asconstructed in the proof of Theorem 3.18. Then, u satisfies u′ < 0 in (0, 1).

Proof. We assume by contradiction that there exists x0 ∈ (0, 1) such that u′(x0) = 0. This zeroof u′ is isolated because otherwise, by reflection and uniqueness for the initial value problem for(2.5), u were even about x0. In view of u′ ≤ 0 on [0, 1] this would imply that u′(x) = 0 for x closeto x0. This, however, is impossible since constants do not solve (2.5).

Then there exist a, b ∈ (0, 1) such that a < x0 < b, u′(a) = u′(b), u′(x) > u′(a) for allx ∈ (a, b). Finally, by choosing a, b close enough to x0 and |u′(a)| small enough we may achievethat (u(b) + u′(b)(a − b))(−u′(b)) ≤ a which will be used to insert a piece of a solution accordingto Theorem 3.18 on [−a, a].

We construct a function v ∈ Nα,β with lower Willmore energy than u and with non-zeroderivative in x0 as follows. v|[b,1] is equal to u|[b,1]. Then v|[a,b) equals the line starting at (b, u(b))with derivative u′(b) and ending at (a, u(b) + u′(b)(a − b)). It remains to define v on [0, a).Here v equals a solution of (1.4) in the interval [−a, a] with boundary values w(±a) = v(a) andw′(a) = u′(a) obtained by a rescaled version of Theorem 3.18. Here we use that, by construction,v(a)(−u′(a)) ≤ a. See Figure 3.

It remains to show that v has strictly lower Willmore energy than u. We first compare theenergies in [−a, a]. Since v(a) > u(a) and v′(a) = u′(a) by a rescaled version of Proposition 3.19we see that the Willmore energy of v|[−a,a] is strictly lower than the Willmore energy of u|[−a,a].

17

Now we compare the energies in [a, b]. From the definition of v and since u′(a) = u′(b) we have

2

b∫

a

κh[u]2ds[u] − 2

b∫

a

κh[v]2ds[v]

= 2

b∫

a

u′′2u

(1 + u′2)5

2

dx + 2

b∫

a

1

u√

1 + u′2 dx + 4

b∫

a

u′′

(1 + u′2)3

2

dx − 2

b∫

a

1

v√

1 + v′2dx

≥ 2

b∫

a

1

u√

1 + u′2 dx − 2

b∫

a

1

v√

1 + v′2dx ≥ 0

where in the last step we used that v(x) ≥ u(x) in [−1, 1] and |v′| ≥ |u′| in [a, b]. Comparing thetotal Willmore energies we then have Wh(u) > Wh(v). A contradiction since u is the minimiserfor Mα,β in Nα,β .

a) a bx0

u

α

u′(x0) = 0

b) a bx0

u

α

1. straight line

c) a bx0

u

α

v2. v|[0,a] solution

according to Theorem 3.18


4 Existence result: The case β < 0

In this section we consider β < 0 fixed, while α varies in the positive real numbers.

The case β < 0 is quite different from β ≥ 0. In the latter, our constructions were based oninserting parts of geodesic circles. Here, also catenoids will play an important role. Each of theseminimal surfaces is generated by the graph of u(x) := cosh(bx)/b, x ∈ [−1, 1], for some b > 0.They are solutions of (1.4) for particular values of α and β. Given β < 0 we denote

αβ :=cosh(b)

bwith b = arsinh(−β). (4.1)

18

Notice that αβ arsinh(−β) =√

1 + β2. We comment on these particular solutions in some moredetail in the next subsection. Then, the cases α > αβ and α < αβ have to be treated separately.In the first case the energy is increasing for α increasing, while in the second case it is decreasingfor α increasing. Moreover, the behaviour of the solution we construct is different in the two cases.If α > αβ the solution satisfies u′ > 0 in (0, 1] while for α < αβ a further critical point couldin principle appear in (0, 1). An intuition for this is given by looking for a function of the kindv(x) = cosh(λ(x − d))/λ choosing λ and d suitably such that v satisfies v(1) = α and v′(1) = −β.If α > αβ , then d < 0. This tell us that, in some sense, there is not enough space for a catenoid.On the other hand, if α < αβ , then d > 0 so there is too much space for a catenoid. One couldthink that a further critical point should show up in (0, 1) together with a solution for β = 0 inthe inner part. By Lemma 5.2 this will certainly happen for α close enough to 0. However, weare not able to determine the precise range of α ∈ (0, αβ) where this extra local minimum may beobserved. The function uCl(x) = 2 −

√2 − x2 for x ∈ [−1, 1] (part of the – projected – Clifford

torus) solves the Willmore equation (1.4) for α = 1 and β = −1 (α < αβ) and has no critical pointin (0, 1).

4.1 The case α = αβ: The catenoid

We summarise the main properties of the catenoids as explicit minimal surface solutions.

Lemma 4.1. For β < 0 and α such that α = cosh(b)/b with b = arsinh(−β), the part of thecatenoid Γ ⊂ R

3 generated by the function u(x) = cosh(bx)/b, x ∈ [−1, 1], solves the Dirichletproblem (1.4).

Moreover, the corresponding surface of revolution is the minimiser of the Willmore functional(1.1) among all axially symmetric surfaces generated by graphs of symmetric positive functions inC1,1([−1, 1], (0,∞)) with v(±1) = α and v′(1) = −β.

Proof. Rotating the graph of u around the x-axis generates a minimal surface, i.e. a surfacesuch that H ≡ 0. Moreover, by the choice of b the function u satisfies the boundary conditionsu(±1) = αβ and u′(1) = −β. For any v ∈ Nα,β we have

Wh(v) = 2

1∫

0

(v′′

(1 + v′2)3

2

− 1

v(x)√

1 + v′(x)2

)2

v(x)√

1 + v′(x)2 dx + 8

1∫

0

v′′

(1 + v′2)3

2

dx

≥ −8β√

1 + β2= Wh(u). (4.2)

This shows that u minimises Mα,β and, by Lemma 2.1, that the axially symmetric surface generatedby u minimises the Willmore functional among axially symmetric surfaces generated by graphs ofsymmetric functions satisfying the prescribed boundary conditions.

Remark 4.2. Notice that given β < 0, there exist unique associated b and αβ defined as in (4.1).When β varies in the negative real numbers, αβ is bounded from below. Indeed, the functionb 7→ 1

b cosh(b) has precisely one minimum at b∗ > 0 which is the solution of the equation

cosh(b∗) = b∗ sinh(b∗), b∗ = 1.1996786 . . . .

The value α∗ defined in (1.5) denotes the minimal value of (0,∞) ∋ b 7→ cosh(b)/b. For α < α∗,

there are no minimal surfaces of revolution solving (1.4). If α = α∗, then cosh(b∗x)b∗ is a minimal

19

surface solution for the boundary datum β = − sinh(b∗). In the case α > α∗, there are two positivereal numbers b1(α), b2(α) such that

b1(α) < b∗ < b2(α) andcosh(b1)

b1= α =

cosh(b2)

b2. (4.3)

Two different minimal surfaces with the same height α in 1 and different boundary slopes corre-spond to these two values. These two catenoids play an important role in what follows.

4.2 The case α > αβ

In this case the height prescribed at the boundary is bigger than the height of the catenoid centeredat 0 and having derivative −β at x = 1. As observed in Remark 4.2, there are two catenoids thatin 1 have the height α. These are cosh(b1x)/b1 and cosh(b2x)/b2 with b1 = b1(α) and b2 = b2(α)defined in (4.3). Since α = cosh(b1)/b1 = cosh(b2)/b2 > αβ = cosh(b)/b, it follows that b1 < b < b2

and sinh(b1) < −β = sinh(b) < sinh(b2). So, close to x = 1, the graph of u ∈ Nα,β is between thegraphs of the two catenoids (see Figure 4). In the following we use this observation to characterisefunctions in Nα,β with low Willmore energy.

α

αβ

cosh(b1x)b1

u ∈ Nα,β

cosh(b2x)b2

cosh(bx)b

α

αβ

sinh(b1)

−β = sinh(b)

−β = sinh(b)

sinh(b2)

Figure 4: Comparison between u ∈ Nα,β and the catenoids cosh(b1x)/b1 and cosh(b2x)/b2.

We explain first how to lower the Willmore energy by inserting C1,1-smoothly suitable parts ofcatenoids. This construction also yields that we may restrict ourselves to functions increasing in[0, 1]. To proceed we have to distinguish between α ≤ −β and α > −β. The line y 7→ αy is crucialfor rescaling and the different positions of curves in Nα,β relative to this line close to x = 1 requiredifferent geometric constructions. If −β ≥ α, these constructions allow for suitably modifyingminimising sequences so that strong enough a priori bounds are available. If −β < α, we need topass to a smaller class of admissible functions instead of Nα,β in order to avoid a possible loss ofcompactness.

4.2.1 First observations

In this subsection we introduce some geometric constructions which lower the Willmore energyand will be used repeatedly in the rest of this section. In the next lemma we formulate a criterion

20

which allows for inserting a piece of a catenoid in a C1,1-smooth way. This criterion is dual to thecondition 0 = x0 + v(x0)v

′(x0) which allows for inserting C1,1-smoothly a part of a geodesic circleon [−x0, x0].

Lemma 4.3. Fix a > 0. Let f ∈ C1,1([0, a], (0, +∞)) be such that f ′(0) = 0. Furthermore assumethat there is x0 ∈ (0, a) such that f ′(x0) > 0 and

1 − 1√1 + f ′(x0)2

cosh

(√1 + f ′(x0)2

f(x0)x0

)= 0. (4.4)

Then, the function

v(x) :=

1

γcosh(γx) for x ∈ [0, x0]

f(x) for x ∈ [x0, a]with γ :=

√1 + f ′(x0)2

f(x0)

is in C1,1([0, a], (0, +∞)), and it satisfies v′(0) = 0.

Proof. It is sufficient to study the behaviour in x0. We see that

limxրx0

v(x) =f(x0)√

1 + f ′(x0)2cosh

(√1 + f ′(x0)2

f(x0)x0

)= f(x0),

using (4.4). For the derivative we find

limxրx0

v′(x) = sinh

(√1 + f ′(x0)2

f(x0)x0

)=

√√√√cosh2

(√1 + f ′(x0)2

f(x0)x0

)− 1 = f ′(x0),

using (4.4) again and the fact that f ′(x0) > 0.

Remark 4.4. Notice that, by the convexity of cosh, if f ′(x) ≤ δ for all x ∈ [0, a] then also v′(x) ≤ δfor all x ∈ [0, a].

In the case β ≥ 0 we could without loss of generality consider only functions satisfying x +u(x)u′(x) ≥ 0 (or ≤ 0). In the next lemma we deduce a dual condition in the case β < 0 andα > αβ . Here we use that an arc of catenoid gives the lowest Willmore energy when connecting onepoint with prescribed positive derivative to another with prescribed and bigger positive derivative.As a consequence we see that without loss of generality it is sufficient to consider functions u ∈ Nα,β

such that u′ > 0 in (0, 1].

Lemma 4.5. For each u ∈ Nα,β there exists v ∈ Nα,β such that Wh(v) ≤ Wh(u), v′ > 0 in (0, 1]and v satisfies

1 − 1√1 + v′(x)2

cosh

(√1 + v′(x)2

v(x)x

)≥ 0 in [0, 1]. (4.5)

Proof. Let u ∈ Nα,β be arbitrary. For easy reference we here denote by g(x) the function on theleft hand side of inequality (4.5) with v replaced by u. Obviously, g(0) = 0. Since b = arsinh(−β),we find in x = 1:

g(1) = 1 − 1√1 + β2

cosh

(√1 + β2

α

)= 1 − 1

cosh(b)cosh

(cosh(b)

α

)

=1

αβ

(αβ − 1

bcosh

(αβ

αb))

=1

bαβ

(cosh(b) − cosh

(αβ

αb))

> 0,

21

using (4.1) and that α > αβ . If g(x) is negative at some point in (0, 1), there exists a largestx0 ∈ (0, 1) such that g(x0) = 0 and g(x) > 0 in (x0, 1]. We first observe that u′(x0) > 0. Thisfollows from u′(1) > 0, the continuity of u′ and the fact that u′(x) 6= 0 for x > 0 where g(x) ≥ 0.Then by Lemma 4.3 with a = 1 and f = u we can define a new function v that coincides with u on[x0, 1] and with a cosh on [0, x0] (see Figure 5, a)). Since v′(0) = 0 we may extend it by symmetryto a C1,1-function on [−1, 1]. For this new function (4.5) is always satisfied. Moreover v has lowerWillmore energy than u. Indeed, we have

Wh(u) = 2

1∫

x0

κh[u]2dsh[u] + 2

x0∫

0

(u′′(x)

(1 + u′(x)2)3

2

− 1

u(x)√

1 + u′(x)2

)2

u(x)√

1 + u′(x)2dx

+8

x0∫

0

u′′(x)

(1 + u′(x)2)3

2

dx ≥ 2

1∫

x0

κh[u]2dsh[u] + 8u′(x0)√

1 + u′(x0)2= Wh(v),

by definition of v and since x 7→ cosh(bx)/b satisfies H(x) ≡ 0.Finally, v′(x) > 0 in (0, 1) since v satisfies (4.5) in [0, 1] and v′(1) = −β > 0.

Remark 4.6. Notice that if u′(x) ≤ δ for all x ∈ [0, 1] then also v′(x) ≤ δ for all x ∈ [0, 1]. Thisis due to the convexity of cosh.

a)0 1

α

catenoid

u

x0b) 0 1r

αα

catenoid

u

u

Figure 5: Proof of Lemma 4.5 (left) and of Lemma 4.11 (right).

In what follows we consider functions having the following property:

u satisfies: 1 − 1√1 + u′(x)2

cosh

(√1 + u′(x)2

u(x)x

)≥ 0 in [0, 1] and u′ > 0 in (0, 1]. (4.6)

We first remark that this condition is scaling invariant, i.e. it is also satisfied for ur(x) = 1ru(rx),

x ∈ [−1r , 1

r ]. The fact that u ∈ Nα,β satisfies (4.6) gives us information on the behaviour of thegraph of u with respect to the two catenoids going through the point (1, α) and centered at theorigin. We recall that these are the functions

cosh(b1x)

b1and

cosh(b2x)

b2with b1 < b∗ < b2 such that

cosh(b1)

b1= α =

cosh(b2)

b2. (4.7)

22

Here b∗ is the unique solution of cosh(b∗) = b∗ sinh(b∗), b∗ = 1.1996786 . . .. We recall that alsob1 < b < b2.

One might expect that x 7→ 1b1

cosh(b1x) and x 7→ 1b2

cosh(b2x) could serve as comparisonfunctions. Unfortunately this works out only partially.

Lemma 4.7. Let u ∈ Nα,β satisfy (4.6). For x ∈ [0, 1) we have:

u(x) =1

b1cosh(b1x) ⇒ u′(x) ≥ sinh(b1x).

More restrictively, if x ∈ [b∗/b2, 1], then

u(x) =1

b2cosh(b2x) ⇒ u′(x) ≤ sinh(b2x).

Proof. It is convenient to rewrite the inequality (4.6) by means of cosh(arsinh(y)) =√

1 + y2 asfollows

arsinh(u′(x)) ≥ cosh(arsinh(u′(x)))x

u(x)in [0, 1],

oru(x)

x≥ cosh(arsinh(u′(x)))

arsinh(u′(x))in (0, 1],

We already know – see Figure 4 – that in a left neighbourhood of 1 we have cosh(b1x)/b1 > u(x) >cosh(b2x)/b2. Let x ∈ (0, 1) be such that u(x) = cosh(bix)/bi, i = 1 or 2. Then from (4.6) itfollows that

cosh(bix)

bix≥ cosh(arsinh(u′(x)))

arsinh(u′(x)). (4.8)

We consider first i = 1. Since x ∈ (0, 1), b1 < b∗ which is the minimum of g(b) = 1b cosh(b), it

follows that b1x is left from this minimum. Hence, an argument with a smaller g-value than b1xmust be right from b1x, i.e. arsinh(u′(x)) ≥ b1x, u′(x) ≥ sinh(b1x).

For i = 2 and x ≥ b∗/b2 we have that b2x is right of the g-minimum b∗. Smaller g-values thang(b2x) are attained at most left from b2x, i.e. if arsinh(u′(x)) ≤ b2x, u′(x) ≤ sinh(b2x).

Remark 4.8. 1. If a function u ∈ Nα,β satisfying (4.6) intersects x 7→ 1b1

cosh(b1x) in a largestpoint x0 ∈ (0, 1), then left of x0, it is below the cosh so that u′(x0) ≤ sinh(b1x0). The previouslemma shows that, on the other hand, u′(x0) ≥ sinh(b1x0) so that u′(x0) = sinh(b1x0). Thefunction u is in x0 tangent to x 7→ 1

b1cosh(b1x) so that the latter may replace C1,1-smoothly

u|[−x0,x0]. This new function v is again in Nα,β , has lower Willmore energy, satisfies (4.6)and in addition

v(x) ≤ cosh(b1x)

b1in [0, 1]. (4.9)

2. Analogously we may achieve on minimising sequences that

u(x) ≥ cosh(b2x)

b2in [b∗/b2, 1].

Unfortunately there is no obvious mechanism to achieve a lower bound also on [−b∗/b2, b∗/b2].

3. Analogously one may also achieve on minimising sequences that v(x) ≥ α∗|x|. Again, thisbound is not sufficient in order to ensure compactness.

In order to prove strong enough lower bounds on suitable minimising sequences, we first achieveuniform derivative bounds. Then, the following lemma will prove to be useful.

23

Lemma 4.9. Let α > 0 and β < 0 be arbitrary and u ∈ Nα,β such that there exists x0 ∈ [0, 1) sothat u′(x0) = 0, u′ > 0 in (x0, 1] and u′ ≤ 0 in [0, x0]. Let γ := maxx∈[x0,1] u

′(x). Then it holdsthat

minx∈[0,1]

u(x) = u(x0) ≥ γ1 − x0

eC − 1,

where C > 0 a constant depending monotonically on Wh(u), γ and β.

Proof. We estimate the Willmore energy from below as follows

Wh(u) ≥ 2

1∫

x0

(u′′u

(1 + u′2)3

2

+1√

1 + u′(x)2

)2 √1 + u′(x)2

u(x)dx

≥ 2

1∫

x0

1

u√

1 + u′(x)2dx + 4

1∫

x0

u′′

(1 + u′2)3

2

dx

= 2

1∫

x0

1

u√

1 + u′(x)2dx − 4

β√1 + β2

.

We recall here that from (4.2) it follows that Wh(v) ≥ −8β/√

1 + β2 for all v ∈ Nα,β . Since u′ ≤ γand, hence, u(x) ≤ u(x0) + γ(x − x0) for x ∈ [x0, 1] we get

Wh(u) ≥ 2√1 + γ2

1∫

x0

1

u(x0) + γ(x − x0)dx − 4

β√1 + β2

=2

γ√

1 + γ2log

(1 +

γ(1 − x0)

u(x0)

)− 4

β√1 + β2

that gives

1 +γ(1 − x0)

u(x0)≤ eC

with C depending monotonically (increasing) on Wh(u), β and γ. The claim follows.

Later we show by possibly inserting a rescaled solution with zero boundary slope that theassumption just made on u ∈ Nα,β is not restrictive.

In the following remark we collect some conclusions which can be drawn for bounds on thederivatives of functions satisfying (4.6).

Remark 4.10. Let u ∈ Nα,β satisfy (4.6) in [0, 1]. Let b1 = b1(α) and b2 = b2(α) be as defined in(4.7). Then, one has the following inequalities:

1. u′(x) = −β implies u(x) ≥ αβx with αβ defined in (4.1);

2. u′(x) = sinh(bi), i = 1, 2, implies u(x) ≥ αx;

3. u(x) ≤ αx implies sinh(b1) ≤ u′(x) ≤ sinh(b2). (In general u(x) ≤ γx, γ > α∗, impliesbounds for the value of the derivative).

As explained at the beginning of this section we proceed now by discussing the cases α ≤ −βand α > −β separately. This distinction requires to study the minimisation process in differentclasses of admissible functions.

24

4.2.2 The case −β ≥ α

We first prove that the energy is monotonically increasing in α. Then, using that u satisfies (4.6),we get a priori bounds on the derivative leading to an existence theorem.

Monotonicity of the optimal energy

For β ≥ 0, when studying monotonicity of the energy we constructed new functions with lowerWillmore energy by inserting a suitable arc of a circle. Here we do an analogous constructioninserting arcs of catenoids.

Lemma 4.11. Assume that u ∈ Nα,β satisfies (4.6). Then for each ∈ (1, α/αβ) there existsu ∈ C1,1([−, ], (0,∞)) positive, symmetric and such that u(±) = α, u′

() = −β, (4.5) issatisfied in [0, ] and u′

> 0 in (0, ] as well as

∫

−


Proof. The idea of the construction is to change the graph of u|[0,r] for some r > 0 by inserting C1,1-smoothly an arc of a catenoid, see Figure 5, b). Then, translating and extending it by symmetrywe find an even function. Choosing r appropriately depending on , we obtain a function definedin the bigger interval [−, ] and satisfying (4.5) in [0, ]. We give now the technical details of theconstruction.

For ∈ (1, α/αβ) let r ∈ (0, 1) be the biggest element in (0, 1) such that ϕ(r) = 0 where ϕ isdefined by

ϕ(x) = x − 1 + − arsinh(u′(x))u(x)√

1 + u′(x)2for x ∈ [0, 1]. (4.10)

Such an element exists since we have ϕ(0) > 0 and, by recalling arsinh(−β)/√

1 + β2 = 1/αβ , that

ϕ(1) < 0. Let λ be defined by λ =√

1 + u′(r)2/u(r). Then the function u defined in [0, 1] by

u(x) :=

u(x + 1 − ) if r − 1 + ≤ x ≤ ,

1λ cosh(λx) if 0 ≤ x < r − 1 + ,

and symmetrically extended to [−1, 1] is in C1,1([−1, 1], (0,∞)). It satisfies u(±) = α, u′() =

−β, u′ > 0 in (0, ] and has a smaller curvature integral than u.

It remains to prove that u satisfies (4.5) in [0, ]. For easy notation let consider ψ defined by

ψ(x) =1√

1 + u′ (x)2cosh

√1 + u′ (x)2

u(x)x

for x ∈ [0, ].

We need to show that ψ(x) ≤ 1 in [0, ]. By construction, ψ(x) ≡ 1 in [0, r − 1 + ] and from√1 + β2/αβ = arsinh(−β) we conclude that ψ() < 1. The claim follows since, by choice of r, the

biggest point in [0, ] such that ψ(x) = 1 is x = r − 1 + .

By rescaling we obtain:

Corollary 4.12. For each u ∈ Nα,β such that u satisfies (4.6) and for each γ ∈ [αβ , α) there existsa positive symmetric v ∈ C1,1([−1, 1], (0,∞)) such that v satisfies (4.6), v(±1) = γ, v′(1) = −βand Wh(v) ≤ Wh(u).

25

Proof. If γ = αβ the claim follows choosing v(x) = cosh(bx)/b with b = −arsinh(−β). If γ ∈ (αβ , α)the claim follows from Lemma 4.11 by rescaling.

Proposition 4.13. Let Mα,β be defined as in (2.7). Then for α ≥ α ≥ αβ we have Mα,β ≥ Mα,β .

Proof. Let (uk)k∈N be a minimising sequence for Mα,β in Nα,β . By Lemma 4.5 for all k ∈ N thereexists vk ∈ Nα,β such that Wh(vk) ≤ Wh(uk) and vk satisfies (4.6). Then Corollary 4.12 yields theclaim.

Properties of minimising sequences

In the next two lemmas we achieve bounds on the derivative. We first observe that we can assumethat when the graph of u is above the line y 7→ αy the derivative cannot be equal to −β. On theother hand, condition (4.6) (in particular (4.5)) gives bounds on the derivatives when the graphof u is below the line y 7→ αy.

Lemma 4.14. Let u ∈ Nα,β satisfy (4.6). Then, there exists v ∈ Nα,β with lower Willmore energythan u satisfying (4.6) and v(x) < αx for all x ∈ (0, 1) with v′(x) = −β.

a) 0 1x0

line y 7→ αy

first pointwithu′ = −β

u

α

b) 0 1x0

u

α

1. wρ, i.e.

u|[0,x0] elongated

c) 0 1x0

u

α

w

2. v, i.e.w rescaled


Proof. Let x0 be the smallest element in [0, 1] such that u′(x0) = −β and u(x0) ≥ αx0. If x0 = 1the claim follows with v = u. If x0 < 1 and u(x0) = αx0 the function v(x) = u(x0x)/x0,x ∈ [−1, 1], yields the claim. On the other hand, if x0 < 1 and u(x0) > αx0 by using a scaledversion of Lemma 4.11 we “extend” the function u|[−x0,x0] by inserting a cosh (see Figure 6). Foreach ∈ (x0, u(x0)/αβ) there exists w ∈ C1,1([−, ], (0,∞)) with lower Willmore energy thanu|[−x0,x0] such that w(±) = u(x0) and w′

() = −β. We then choose = u(x0)/α and v to beequal to the function w rescaled to the interval [−1, 1]. The choice of is such that we extendu|[0,x0] until we touch the line y 7→ αy. Notice that v ∈ Nα,β and that v satisfies (4.6). It remainsto check that if v′(x) = −β for some x ∈ (0, 1) then v(x) < αx. By construction, the function w

is given by

w(x) :=

u(x + x0 − ) if r − x0 + ≤ x ≤ ,

1

λcosh(λx) if 0 ≤ x < r − x0 + ,

26

for some r ∈ (0, x0) and λ = λ(r) > 0. If there exists x ∈ (0, 1) such that v′(x) = −β thenw′

(x) = −β. If x ∈ [0, r − x0 + ] then w′(x) = −β implies λx = b and hence

v(x)

x=

w(x)

x=

cosh(λx)

λx= αβ < α.

If instead x ∈ (r +−x0, ] then u′(x+x0−) = −β and hence u(x+x0−) < α(x+x0 −)from which it follows

v(x)

x=

u(x + x0 − )

x=

u(x + x0 − )

x + x0 −

x + x0 −

x< α.

The claim follows.

In the proof we use only that α > αβ .We recall the definition (4.7) of the positive real numbers b2 = b2(α) and b1 = b1(α) such

that cosh(b2)/b2 = α = cosh(b1)/b1 and b2 > b∗ > b1, with b∗ being the solution of cosh(b∗) =b∗ sinh(b∗).

Lemma 4.15. Let u ∈ Nα,β satisfy (4.6) and u′(x) 6= −β for all x ∈ (0, 1) with u(x) ≥ αx. Then,u′(x) ≤ sinh(b2) in [0, 1].

Proof. We assume first in addition that there exists a left neighbourhood of 1 such that there wehave u(x) ≤ αx. Such a neighbourhood always exists if −β > α. By 3) in Remark 4.10 we havethat u′(x) ≤ sinh(b2) for all x such that u(x) ≤ αx. Hence, when the graph of u is below theline y 7→ αy we have a bound for the derivative. Let now x ∈ (0, 1) be such that u(x) > αx. Weshow that u′(x) < −β. Let x0 be the smallest element in (x, 1) such that u(x0) = αx0. Thenu′(x0) ≤ α ≤ −β. If we assume that u′(x) ≥ −β then, by continuity, there exists y ∈ [x, x0] suchthat u′(y) = −β. A contradiction. Hence u′(x) < −β < sinh(b2).

It remains to consider the case where there is no left neighbourhood of 1 such that there wehave u(x) ≤ αx. Then, necessarily −β = α, and we have a sequence xk ր 1 with u(xk) > αxk.Looking at the first point right from xk, where u reaches y 7→ αy shows that u(x) > αx on[0, xk]. Otherwise the mean value theorem would yield a point ξ ∈ (0, 1) with u(ξ) ≥ αξ andu′(ξ) = α = −β, a contradiction. Letting k → ∞ yields u(x) > αx on [0, 1). By u′(0) = 0 weconclude that 0 ≤ u′(x) < −β < sinh(b2) on [0, 1) in this case.


Proposition 4.16. Let (uk)k∈N be a minimising sequence for Mα,β in Nα,β such that Wh(uk) ≤Mα,β + 1 for all k ∈ N. Let b2 = b2(α) and b1 = b1(α) be as defined in (4.7). Then, there existsa minimising sequence (vk)k∈N ⊂ Nα,β such that for all k ∈ N the function vk satisfies (4.6),Wh(vk) ≤ Wh(uk),

sinh(b2) ≥ v′k(x) > 0 for all x ∈ (0, 1] and Cα,β ≤ vk(x) ≤ 1

b1cosh(b1x) in [−1, 1], (4.11)

with a constant Cα,β > 0 depending on Mα,β, sinh(b2) and −β.

Proof. By Lemmas 4.5, 4.14 and 4.15 for each uk there exists vk ∈ Nα,β with lower Willmore energythan uk such that vk satisfies (4.6) and sinh(b2) ≥ v′k > 0 in (0, 1]. According to Remark 4.8 we mayalso achieve that vk(x) ≤ 1

b1cosh(b1x). The estimate from below for vk follows from Lemma 4.9.

The constant Cα,β denotes the term on the right hand side of Lemma 4.9 with γ = sinh(b2) andx0 = 0. Notice that by the assumption Wh(uk) ≤ Mα,β + 1 the constant Cα,β depends only onMα,β , sinh(b2) and −β.

27

Proof of the existence theorem

The real numbers b2 = b2(α) and b1 = b1(α) are defined in (4.7) and such that cosh(b2)/b2 = α =cosh(b1)/b1 and b2 ≥ b∗ ≥ b1, with b∗ the solution of cosh(b∗) = b∗ sinh(b∗).

Theorem 4.17 (Existence and regularity). For β < 0 and α such that α > αβ and −β ≥ α thereexists a function u ∈ C∞([−1, 1], (0,∞)) such that the corresponding surface of revolution Γ ⊂ R

3

solves the Dirichlet problem (1.4). This solution is positive and symmetric, and it has the followingproperties:

sinh(b2) ≥ u′(x) > 0 in (0, 1] and1

b1cosh(b1x) ≥ u(x) ≥ Cα,β in [−1, 1]

with a constant Cα,β > 0 depending on Mα,β, sinh(b2) and −β.

Proof. Let (uk)k∈N ∈ Nα,β be a minimising sequence for Mα,β such that Wh(uk) ≤ Mα,β + 1 forall k ∈ N. By Proposition 4.16 we may assume that each element uk of the minimising sequencesatisfies (4.6) and (4.11). The rest of the proof is on the same line as that of Theorem 3.11. Noticethat since uk satisfies (4.6) for all k ∈ N then also u satisfies (4.6) and so u′ > 0 in (0, 1].

We can improve Proposition 4.13 by showing that the energy is strictly increasing in α.

Proposition 4.18. Let Mα,β be defined as in (2.7) and α > αβ. Then, for −β ≥ α > α we haveMα,β > Mα,β .

Proof. Let u ∈ Neα,β be a solution of (1.4) with boundary values α, β as constructed in Theo-rem 4.17. Then Wh(u) = Meα,β and u satisfies (4.6). We first notice that u satisfies (4.6) with astrict inequality. Indeed, if there exists x0 ∈ (0, 1) such that

1 − 1√1 + u′(x0)2

cosh

(√1 + u′(x0)2

u(x0)x0

)= 0,

reasoning as in Lemma 4.5 and using that u is the minimiser in Neα,β , it follows that u|[−x0,x0] isequal to a catenoid. Then, u being a solution of (2.5), it follows that u is a catenoid in [−1, 1].This is not possible since α > αβ . Hence, applying the procedure of Corollary 4.12 to a minimiseru ∈ Neα,β yields a v ∈ Nα,β with strictly lower energy Wh(u) > Wh(v). Since Wh(v) ≥ Mα,β theclaim follows. Notice that the same reasoning shows that also this last inequality is strict.

4.2.3 The case −β < α

In this case we are not able to obtain a bound from above for the derivative for minimising sequencesin Nα,β . Possibly, a loss of compactness could occur. To avoid this problem we restrict the set onwhich we minimise by adding a constraint. We require the derivative to be bounded by α.

We consider

Nα,β = u ∈ Nα,β : u′(x) ≤ α for all x ∈ [0, 1],

and

Mα,β = infu∈ eNα,β

Wh(u). (4.12)

The assumption −β < α ensures that Nα,β is not empty.

Also in this case, by Lemma 4.5 and the subsequent remark, it is sufficient to consider functionsu ∈ Nα,β satisfying (4.6).

28

Monotonicity of the optimal energy

Proceeding as in the previous section we find:

Lemma 4.19. Assume that u ∈ Nα,β satisfies (4.6). Then for each ∈ (1, α/αβ) there existsu ∈ C1,1([−, ], (0,∞)) positive and symmetric such that u(±) = α, u′

() = −β, (4.5) issatisfied in [0, ] and α ≥ u′

> 0 in (0, ] as well as

∫

−


Proof. The proof is exactly as in Lemma 4.11. By the explicit expression of u and the convexityof cosh, we see that u′ ≤ α implies u′

≤ α.

Corollary 4.20. For each u ∈ Nα,β such that u satisfies (4.6) and for each γ ∈ [αβ , α) there existsa positive symmetric v ∈ C1,1([−1, 1], (0,∞)) such that v satisfies (4.6), v(±1) = γ, v′(1) = −β,α ≥ v′(x) > 0 for x ∈ (0, 1] and Wh(v) ≤ Wh(u).

Proposition 4.21. Let Mα,β be defined as in (4.12). Then for α ≥ α we have Mα,β ≥ Mα,β .


The following lemma is the analogue of Lemma 4.14 in the case α ≤ −β.

Lemma 4.22. Let u ∈ Nα,β satisfy (4.6). Then there exists v ∈ Nα,β with lower Willmore energythan u satisfying (4.6) and v(x) > αx as well as v′(x) < −β in (0, 1).

Proof. We first notice that u(x) > αx in (0, 1). If u′(x) < −β in (0, 1) then the claim followswith v = u. Otherwise let x0 ∈ (0, 1) be the smallest element in (0, 1) such that u′(x0) = −β.We repeat then the construction in Lemma 4.14. By using a scaled version of Lemma 4.19 weelongate the function u|[−x0,x0] by inserting a cosh. For each ∈ (x0, u(x0)/αβ) there existsw ∈ C1,1([−, ], (0,∞)) with lower Willmore energy than u|[−x0,x0] such that w(±) = u(x0)and w′

() = −β. We then choose = u(x0)/α and v to be equal to the function w rescaled tothe interval [−1, 1]. The choice of is such that we extend u|[0,x0] until we touch the line y 7→ αy.Notice that v ∈ Nα,β , v satisfies (4.6) and that by convexity of cosh we have v′(x) < −β in [0, 1).

In particular v ∈ Nα,β . Compared with Lemma 4.14, the proof in this case is simpler since weare always above the line y 7→ αy. In [0, x0) there are no points with derivative ≥ −β. With thisconstruction we do not add such points.


Proposition 4.23. Let (uk)k∈N be a minimising sequence for Mα,β in Nα,β such that Wh(uk) ≤Mα,β +1 for all k ∈ N. Let b1 = b1(α) as defined in (4.7). Then, there exists a minimising sequence

(vk)k∈N ⊂ Nα,β such that for all k ∈ N, vk satisfies (4.6), Wh(vk) ≤ Wh(uk) and

−β ≥ v′k(x) > 0 for all x ∈ (0, 1] and 0 < α + β ≤ vk(x) ≤ 1

b1cosh(b1x) in [−1, 1]. (4.13)

Proof. By Lemmas 4.5 and 4.22 for each uk there exists vk ∈ Nα,β with lower Willmore energythan uk such that vk satisfies (4.6) and −β ≥ v′k > 0 in (0, 1]. According to Remark 4.8 we mayalso achieve that vk(x) ≤ 1

b1cosh(b1x). The estimate from below of vk follows directly.

29


We recall that b1 = b1(α) is defined in (4.7) and it is such that cosh(b1)/b1 = α and b1 ≤ b∗, withb∗ solution of cosh(b∗) = b∗ sinh(b∗).

Theorem 4.24 (Existence and regularity). For β < 0 and α such that α > αβ and α > −β thereexists a function u ∈ C∞([−1, 1], (0,∞)) such that the corresponding surface of revolution Γ ⊂ R

3

solves the Dirichlet problem (1.4). This solution is positive and symmetric, and it has the followingproperties:

−β ≥ u′(x) > 0 in (0, 1] and1

b1cosh(b1x) ≥ u(x) ≥ α + β in [−1, 1]. (4.14)

Proof. As in the proof of Theorem 3.11 we find a minimiser u ∈ Nα,β of Wh. This means that uminimises Wh in the class of all positive and symmetric H2(−1, 1)-functions v satisfying v(±1) = α,v′(+1) = −β, and having first derivative bounded pointwise by α. Moreover, since the elements ofthe minimising sequence satisfy (4.6) and (4.13) then u satisfies also (4.6) and hence (4.14). Sinceu′(x) ≤ −β < α for x ∈ [0, 1], then for |t| sufficiently small u + tϕ ∈ Nα,β for ϕ ∈ H2(−1, 1) with

ϕ(±1) = 0 = ϕ′(±1). Therefore, u is an interior point of Nα,β in H2(−1, 1) and u weakly solves(2.5).

The proof of smoothness of the solution is as in [DDG, Theorem 3.9, Step 2].

Proceeding as in the proof of Proposition 4.18 one can show that the energy also in this caseis strictly increasing in α.

Proposition 4.25. Let −β < α and α > αβ. Let Mα,β be as defined in (4.12). Then Meα,β > Mα,β

for all α such that α > α.

4.3 The case α < αβ

In this case the height prescribed at the boundary is smaller than the height of the catenoidcentered at 0 and having derivative −β at x = 1. This case is not simply the dual of the caseα > αβ . The function b 7→ cosh(b)/b has a unique minimum at b∗ = 1.1996786 . . . and its minimalvalue is α∗ = 1.5088795 . . . (see (1.5)). Hence, when considering α < αβ we have to consider twodifferent cases: when α ≥ α∗ and when α < α∗. The first case is, in some sense, the dual to thecase α > αβ . The constructions and the methods of proof are similar. On the other hand, thecase α < α∗ is completely different. Here, only parts of the functions u ∈ Nα,β close to x = 1 canbe compared with catenoids.

It is useful to restrict further the functions we consider. This restriction is technically importantfor the case α < α∗ but, for the sake of a uniform presentation, we use it in the entire section. Werestrict our study to, what we call, admissible functions. These admissible functions are, however,dense in the space of all symmetric H2((−1, 1), (0,∞))-functions and so, one can stick to them inminimising the Willmore functional without any loss of generality.

Definition 4.26 (Admissible functions). A function u ∈ C1,1([−a, a], (0,∞)), a > 0, is calledadmissible if it is positive, symmetric and if there exist finitely many points 0 = x0 < x1 < x2 <· · · < xm = a such that u|[xj ,xj+1], j = 0, . . . , m−1, is a polynomial of degree at least two, or equalto cosh(λ(x − d))/λ for some λ ∈ (0,∞), d ∈ R, or an arc of a circle with centre on the x-axis oran arc of a solution of (1.4) with β = 0 as constructed in Theorem 3.18.

In what follows, we only perform constructions for admissible functions which yield againadmissible functions. In most cases the starting point will be polynomials.

30

In this section we first show that it is sufficient to consider functions having at most onecritical point in (0, 1) and satisfying a condition dual to (4.6). Moreover, we employ the catenoidsas comparison functions. We then show that the energy is monotonically decreasing in α. Toproceed we need to distinguish the cases α ≥ α∗ and α < α∗. We explain later the strategy ofproof in the two cases.

4.3.1 First observations

The next two lemmas correspond to Lemma 4.5 for α > αβ . There we could restrict ourselvesto functions satisfying u′ > 0 in (0, 1). Here we show that we can restrict ourselves to functionshaving at most one critical point in (0, 1).

Lemma 4.27. Let u ∈ Nα,β be an admissible function in the sense of Definition 4.26. Then, thereexists an admissible function v ∈ Nα,β with lower Willmore energy than u and having at most onecritical point in (0, 1), i.e. either v′ > 0 in (0, 1] or there exists x0 ∈ (0, 1) such that v′(x0) = 0,v′ > 0 in (x0, 1] and v′ < 0 in (0, x0).

Proof. If u does not satisfy u′ > 0 in (0, 1] there exists x0 ∈ (0, 1) such that u′(x0) = 0 andu′(x) > 0 in (x0, 1]. We then replace u|[−x0,x0] with an appropriately rescaled solution of (1.4) withboundary data u(x0) and 0 as constructed in Theorem 3.18. This rescaled function has strictlynegative derivative in (0, x0) by Lemma 3.20. The obtained function v yields the claim.

In the next result we give a condition corresponding to (4.5) in this case. Here we use bothcatenoids and geodesic circles. The condition x + u(x)u′(x) ≥ 0 was a consequence of (4.5) forα > αβ but this is not the case here.

Lemma 4.28. Let u ∈ Nα,β be an admissible function in the sense of Definition 4.26. Then, thereexists an admissible function v ∈ Nα,β and x0 ∈ [0, 1) with v′ < 0 in (0, x0), v′(x0) = 0, v′ > 0 in(x0, 1] and Wh(v) ≤ Wh(u). Moreover, v satisfies

1 − 1√1 + v′(x)2

cosh

(√1 + v′(x)2

v(x)x

)≤ 0 in [x0, 1] (4.15)

and

x + v(x)v′(x) ≥ 0 in [0, 1]. (4.16)

Proof. Let w ∈ Nα,β be the function constructed in Lemma 4.27. One has Wh(w) ≤ Wh(u). Wedenote by x1 ∈ [0, 1) the point such that w′ > 0 in (x1, 1], w′(x1) = 0 and w′ < 0 in (0, x1).

This function w satisfies (4.15) in x = 1 and in x1. Indeed, in x = 1 we find

1 − 1√1 + β2

cosh

(√1 + β2

α

)=

1

αβ

(αβ − 1

bcosh

(αβ

αb))

< 0

since β = − sinh(b), αβ = 1b cosh(b), using (4.1) and α < αβ . In x1 we get 1− cosh(x1/w(x1)) ≤ 0

and equality holds only if x1 = 0. If w satisfies (4.15) in [x1, 1] we define in [0, 1] the functionh(x) := x+w(x)w′(x). The function h is strictly positive in [x1, 1] since w′ ≥ 0 in [x1, 1]. Moreover,h(0) = 0. If h > 0 in (0, x1] the claim follows with v = w and x0 = x1. Otherwise there exists abiggest element x ∈ (0, x1) with h(x) = 0. Then we may substitute w in [−x, x] in a C1,1-smoothway by an arc of a circle lowering the Willmore energy. This new function gives the claim.

It remains to treat the case when w does not satisfy (4.15) in [x1, 1]. For easy reference wedenote here by g the function on the left hand side of (4.15) with v replaced by w. Let x2 be the

31

biggest element in [x1, 1] such that g(x2) = 0 and g(x) ≤ 0 in [x2, 1]. By Lemma 4.3 with a = 1and f = w we can define a new function v coinciding with u on [x2, 1] and being a cosh on [0, x2].Since v′(0) = 0 we may extend it by symmetry to a C1,1-function on [−1, 1]. This new functionalways satisfies (4.15) and has lower Willmore energy. Notice that in this case x0 = 0 and hence,(4.16) is certainly satisfied in [0, 1].

In what follows we consider only admissible functions u ∈ Nα,β satisfying the following condi-tions.

There exists x0 ∈ [0, 1) such that u′ > 0 in (x0, 1], u′(x0) = 0, u′ < 0 in (0, x0),

1 − 1√1 + u′(x)2

cosh

(√1 + u′(x)2

u(x)x

)≤ 0 in [x0, 1],

and x + u(x)u′(x) ≥ 0 in [0, 1].

(4.17)

Before proceeding by proving monotonicity of the energy, we first compare functions in Nα,β

with arcs of catenoids. In the next lemma we show that without loss of generality we may assumethat functions satisfying (4.17) with x0 > 0 satisfies also an uniform bound from below for u(x0)/x0.

Lemma 4.29. Let u ∈ Nα,β be an admissible function in the sense of Definition 4.26 satisfying(4.17) for some x0 > 0. Then, there exists an admissible function v ∈ Nα,β such that Wh(u) ≥Wh(v), v satisfies (4.17) for some x1 > 0 and v(x1) ≥ α

arsinh(−β)(αβ−α)x1.

Proof. We recall that b = arsinh(−β). If u(x0) ≥ αb(αβ−α)x0 the claim follows with v = u. Other-

wise we can construct a function satisfying the claim and with lower Willmore energy than u. Weconsider, starting from 1 and going towards 0, the arc of the catenoid going through (1, α) andhaving derivative −β in 1. This is x 7→ α cosh(bαβ(x− 1 + α/αβ)/α)/bαβ . We follow the catenoidup to its minimum. Since α < αβ , the minimum is achieved in the point 1 − α/αβ ∈ (0, 1). Inthis point, we attach to this catenoid a suitably rescaled solution of (1.4) as constructed in Theo-rem 3.18 with boundary data αb/αβ and 0. Finally, extending the graph by symmetry, we obtaina function v. Notice that v satisfies (4.17) with x1 = 1−α/αβ and that v(x1) = α

b(αβ−α)x1. By the

monotonicity property of the energy in the case β = 0 (see Proposition 3.19) one sees that v haslower Willmore energy than u. Notice that the cosh-part has the lowest possible energy among allcurves connecting the boundary point α with slope −β and any point with horizontal tangent.

The next lemma corresponds to Remark 4.10 in the case α > αβ . We recall here that forα′ ≥ α∗, the real numbers b2 = b2(α

′) and b1 = b1(α′) are defined in (4.7) by cosh(b2)/b2 = α′ =

cosh(b1)/b1 and b2 ≥ b∗ ≥ b1, where b∗ is the solution of cosh(b∗) = b∗ sinh(b∗).

Lemma 4.30. Let u ∈ Nα,β be an admissible function in the sense of Definition 4.26 and let usatisfy (4.17) for some x0 ∈ [0, 1). Consider α′ ≥ α∗. Then for all x ∈ (x0, 1] such that u(x) > α′xwe have that either u′(x) < sinh(b1(α

′)) or u′(x) > sinh(b2(α′)).

Proof. The claim follows directly using that u satisfies in particular (4.15).

The previous lemma shows that when the graph of u ∈ Nα,β is above the line y 7→ α′y, α′ > α∗,we have a bound on the derivative. For this reason it is natural to distinguish below the casesα ≥ α∗ and α < α∗.

32

a)x0

u

α

αb(αβ−α)

u(x0)x0

b) x0

u

α

αb(αβ−α)

u(x0)x0

minimiser

for β = 0

piece of

catenoid

v

Figure 7: The construction in Lemma 4.29. On the left: a function u ∈ Nα,β with u(x0) <αx0/(b(αβ − α)). On the right: a function v ∈ Nα,β with lower Willmore energy than u satisfyingv(x1) = αx1/(b(αβ − α)) with x1 = 1 − α/αβ .


We prove here that Mα,β decreases when α increases to αβ . For later use we work in a more generalsetting.

We start by showing that we can construct functions defined in a smaller interval, with thesame boundary values and with lower Willmore energy. We first prove the result for functionssatisfying (4.17) with x0 = 0 and then extend it to the general case.

Lemma 4.31. Fix t < 0. Assume that u ∈ C1,1([−1, 1], (0,∞)) is an admissible function inthe sense of Definition 4.26 satisfying u(1) < αt, u′(1) = −t and (4.17) with x0 = 0. Thenfor each ∈ (u(1)/αt, 1) there exists an admissible function u ∈ C1,1([−, ], (0,∞)) such thatu(±) = u(1), u′

() = −t, u′ > 0 in (0, ], u satisfies (4.15) in [0, ] (with x0 = 0) as well as

∫

−


Proof. One uses the same construction as in Lemma 4.11. Since α < αβ , this procedure nowshortens the original function.

Lemma 4.32. Fix t < 0. Assume that u ∈ C1,1([−1, 1], (0,∞)) is an admissible function in thesense of Definition 4.26 satisfying u′(1) = −t, u(1) < αt and (4.17) for some x0 ∈ [0, 1).

Then for each ∈ (u(1)/αt, 1) there exists an admissible function u ∈ C1,1([−, ], (0,∞))such that u(±) = u(1), u′

() = −t. Moreover, there exists an x1 ∈ [0, ) with u′(x1) = 0, u′ > 0

in (x1, ] and u′ < 0 in (0, x1), u satisfies (4.15) in [x1, ], x + u(x)u′

(x) ≥ 0 in [0, ] as well as

∫

−


Proof. It combines the constructions of Lemmas 3.3 and 3.16 (inserting circular arcs) and thoseof Lemma 4.31 (inserting catenoidal parts). See Figure 8. We emphasise that these constructionspreserve the strict inequalities for the derivatives. The additional properties of u are ensured by

33

possibly inserting once more a circular arc or a cosh, respectively, into the shortened function.Observe that (4.15) is certainly satisfied in x = for any shortened function.

Notice that if u(x0) ≥ µx0 then also u(x1) ≥ µx1.

a) 0 1c(r) x0

u

r

αCase > 1 − x0:

insert an arc of a

geodesic circle

b) 0 1x0

u

r

αCase < 1 − x0:

insert an arc of a catenoid


Corollary 4.33. Assume that u ∈ Nα,β satisfies (4.17) for some x0 ∈ [0, 1) and that u is anadmissible function in the sense of Definition 4.26. Then for all γ ∈ (α, αβ ] we find an admissiblefunction v ∈ C1,1([−1, 1], (0,∞)) satisfying (4.17) for some x1 ∈ [0, 1), v(±1) = γ, v′(1) = −β aswell as Wh(v) ≤ Wh(u).

Proof. The claim follows from Lemma 4.32 by rescaling and taking t = β and u(1) = α.

Before showing monotonicity of the optimal Willmore energy, we prove a result being dual toLemma 4.14.

Lemma 4.34. Let u ∈ Nα,β be an admissible function in the sense of Definition 4.26 satisfying(4.17) for some x0 ∈ [0, 1). Then, there exists an admissible function v ∈ Nα,β with lower Willmoreenergy than u satisfying (4.17) for some x1 ∈ [0, 1) and v(x) > αx for all x ∈ (0, 1) with v′(x) =−β.

Proof. Let x be the smallest element in [0, 1] such that u′(x) = −β and u(x) ≤ αx. If x = 1 theclaim follows with v = u and x1 = x0. If x < 1 and u(x) = αx the function v(x) = u(xx)/x,x ∈ [−1, 1], yields the claim with x1 = x0/x. Finally, if x < 1 and u(x) < αx by using a scaledversion of Lemma 4.32 we shorten the function u|[−x,x] by inserting a cosh or an arc of a circle.For each ∈ (u(x)/αβ , x) there exists w ∈ C1,1([−, ], (0,∞)) with lower Willmore energy thanu|[−x,x] such that w(±) = u(x) and w′

() = −β. We then choose = u(x)/α such that thegraph of u|[−x,x] is shortened until we touch the line y 7→ αy. We define v to be equal to thefunction w rescaled to the interval [−1, 1]. Notice that v ∈ Nα,β is an admissible function andsatisfies (4.17) for some x1 ∈ [0, 1). It remains to check that if v′(x) = −β for some x ∈ (0, 1) thenv(x) > αx. The function w is given by

w(x) :=

u(x + x − ) if r ≤ x ≤ ,g(x) if 0 ≤ x < r,

for some r ∈ (0, ), and either g(x) = cosh(λx)/λ for some λ > 0 or g is an arc of a geodesiccircle and g′ ≤ 0. If there exists x ∈ (0, 1) such that v′(x) = −β then w′

(x) = −β. Then, either

34

x ∈ [r, ) and w(x) = u(x+ x− ), or x ∈ [0, r) and w(x) = cosh(λx)/λ. In the first case,since x + x − < x and x is the smallest element such that u′(x) = −β and u(x) ≤ αx, thenw(x) = u(x + x − ) > α(x + x − ) and so, by < x

v(x)

x=

w(x)

x=

u(x + x − )

x>

α(x + x − )

x> α.

In the second case, if w′(x) = −β then necessarily λx = b and so

v(x)

x=

w(x)

x=

cosh(λx)

λx= αβ > α.

The claim follows.

Note that u(x0) ≥ µx0 implies that v(x1) ≥ µx1.

a) 0 1x

u

α

Firstpointwithu′ = −β

b) 0 1x

u

α

1. w, i.e.u|[0,x] shortened

c) 0 1x

u

α

2. v, i.e.w rescaled

1. w

Figure 9: The three main steps of the proof of Lemma 4.34.

Proposition 4.35. Let Mα,β be defined as in (2.7) and α < αβ. Then, for α ≤ α we haveMα,β ≥ Mα,β .

Proof. By density, we may choose a minimising sequence (uk)k∈N ⊂ Nα,β for Mα,β consisting ofpositive symmetric polynomials of degree at least two. These functions are in particular admissiblein the sense of Definition 4.26. Then, by Lemma 4.28 there exists a sequence (vk)k∈N ⊂ Nα,β ofadmissible functions such that vk satisfies (4.17) for some xk ∈ [0, 1) and Wh(vk) ≤ Wh(uk).Corollary 4.33 then yields the claim.

4.3.3 The case α ≥ α∗

In this case we can compare u ∈ Nα,β with the catenoids centered at 0 and going through (1, α).As observed in Remark 4.2, these are the functions x 7→ cosh(b1x)/b1 and x 7→ cosh(b2x)/b2 withb1 = b1(α) and b2 = b2(α) the positive real numbers such that b1 ≤ b∗ ≤ b2 and cosh(b1)/b1 =α = cosh(b2)/b2, with b∗ the solution of cosh(b∗) = b∗ sinh(b∗). Since α < αβ one sees that or

35

a)

α∗

αβCase −β > sinh(b2):

cosh(bx)b

cosh(b1x)b1

cosh(b2x)b2

α

u ∈ Nα,β b)

α∗

αβ

cosh(bx)b

cosh(b1x)b1

cosh(b2x)b2

αu ∈ Nα,β

Case −β < sinh(b1):

Figure 10: Two possibilities for the behaviour near 1 of the graph of u ∈ Nα,β when α∗ ≤ α < αβ .Compare with Figure 4.

−β < sinh(b1) or −β > sinh(b2) (see Figure 10). Notice that since sinh(b1) ≤ α ≤ sinh(b2), thesetwo cases correspond to the two cases −β < α and −β ≥ α that we have treated separately alsoin the case α > αβ .

By comparing u ∈ Nα,β with the catenoids we show that in the case −β < sinh(b1) ≤ α it issufficient to consider functions u ∈ Nα,β such that u(x) ≥ cosh(b1x)/b1, i.e. remaining above thelarger of the two catenoids. This in particular implies u(x) ≥ αx. This together with Lemma 4.30gives bounds on the derivative. In the case −β > sinh(b2) ≥ α we first prove bounds on thederivative using Lemmas 4.30 and 4.34. Then, by Lemma 4.9 we get a bound from below for thefunction.


In the next lemmas it is convenient to distinguish the cases −β < sinh(b1) and −β > sinh(b2)because of the different behaviour with respect to the line y 7→ αy. Recall that b1 = b1(α) andb2 = b2(α) are the positive real numbers such that b1 ≤ b∗ ≤ b2 and cosh(b1)/b1 = α = cosh(b2)/b2,with b∗ being the solution of cosh(b∗) = b∗ sinh(b∗).

Lemma 4.36. We assume in addition that −β < sinh(b1). Let u ∈ Nα,β be an admissible functionin the sense of Definition 4.26. Assume furthermore that (4.17) is satisfied for some x0 ∈ [0, 1) andthat u′(x) 6= −β for all x ∈ [0, 1) with u(x) ≤ αx. Then, u(x) ≥ cosh(b1x)/b1 and u′(x) ≤ sinh(b1x)in [0, 1).

Proof. Since −β < sinh(b1) ≤ α, u(x) > cosh(b1x)/b1 in a left neighbourhood of 1. Moreover,since u satisfies (4.17) (in particular (4.15) in [x0, 1]) one sees as long as u(x) ≥ cosh(b1x)/b1 in[x0, 1] that

cosh(b1x)

b1x≤ u(x)

x≤ cosh(arsinh(u′(x)))

arsinh(u′(x)).

Since arsinh(u′(1)) < b1 ≤ b∗, we conclude by continuity that u′(x) ≤ sinh(b1x). Hence, u(x) >1b1

cosh(b1x) on [x0, 1) and so, also in [0, 1).

Lemma 4.37. We assume in addition that −β > sinh(b2). Let u ∈ Nα,β be an admissible functionin the sense of Definition 4.26. Assume furthermore that (4.17) is satisfied for some x0 ∈ [0, 1)and that u′(x) 6= −β for all x with u(x) ≤ αx. Then, there exists an admissible function v ∈ Nα,β

36

with lower Willmore energy than u, which satisfies (4.17) for some x1 ∈ [0, x0] and v′(x) ≤ −β in[0, 1].

Proof. Since −β > sinh(b2) ≥ cosh(b2)b2

= α, u(x) < αx in a left neighbourhood of 1. Let x1 bethe biggest element in (0, 1) such that u(x1) = αx1 and u′(x1) ≤ α. Such an element exists sinceu(0) > 0. Since u′(x1) ≤ α < −β and u′ 6= −β in (x1, 1) we have u′ ≤ −β in [x1, 1].

If u′(x1) < α then u′(x) < α also in a left neighbourhood of x1 and u(x) > αx in this

neighbourhood. Hence by Lemma 4.30 for these points u′(x) < sinh(b1). Since sinh(b1) ≤ cosh(b1)b1

=α, then, by continuity, u(x) > αx and u′(x) ≤ sinh(b1) ≤ −β for all x ∈ [0, x1].

From Lemma 4.30 it follows also that u′(x1) = α can hold only if α = α∗ since sinh(b1) < α <sinh(b2) for α 6= α∗. Hence for α > α∗ the claim is proved with u = v. If α = α∗ and u′(x1) = α∗

then we substitute u in [−x1, x1] by the function cosh(λx)/λ with λ = b∗/x1 and b∗ defined in(1.5). We get a new function v ∈ Nα,β with lower Willmore energy than u and such that v satisfies(4.17) with x1 = 0, v′(x) ≤ α∗ and v(x) ≥ α∗x in [0, x1].


Proposition 4.38. Let (uk)k∈N be a minimising sequence for Mα,β of admissible functions in thesense of Definition 4.26 in Nα,β such that Wh(uk) ≤ Mα,β + 1 for all k ∈ N. Then, there exists aminimising sequence (vk)k∈N ⊂ Nα,β of admissible functions satisfying (4.17), Wh(vk) ≤ Wh(uk)and

max−β, sinh(b1)

≥ v′k(x) ≥ −arsinh(−β)

α(αβ−α) and Cα,β ≤ vk(x) ≤

√1 + α2 − x2, (4.18)

in [0, 1] with a constant Cα,β > 0 depending on α, −β and Mα,β.

Proof. By Lemmas 4.28, 4.34, 4.36 if −β < sinh(b1) or 4.37 if −β > sinh(b2), and Lemma 4.29 foreach uk there exists vk ∈ Nα,β with lower Willmore energy than uk such that vk satisfies (4.17) forsome xk ∈ [0, 1) and

v′k(x) ≤ max−β, sinh(b1) and vk(xk) ≥α

arsinh(−β)(αβ − α)xk. (4.19)

Since vk satisfies (4.16) we get vk(x) ≤√

1 + α2 − x2 in [0, 1] and

v′k(x) ≥ − x

vk(x)≥ − xk

vk(xk)≥ −arsinh(−β)

αβ − α

αfor x ∈ [0, xk],

while v′k ≥ 0 in [xk, 1]. The estimate from below for vk follows from the second estimate in (4.19)if xk ≥ 1/2 and from Lemma 4.9 if xk ≤ 1/2.


We recall here that for α ≥ α∗, b1 = b1(α) denotes the positive real number such that cosh(b1)/b1 =α and b1 ≤ b∗ with b∗ being the solution of cosh(b∗) = b∗ sinh(b∗).

Theorem 4.39 (Existence and regularity). For β < 0 and α such that α∗ ≤ α < αβ there existsa function u ∈ C∞([−1, 1], (0,∞)) such that the corresponding surface of revolution Γ ⊂ R

3 solvesthe Dirichlet problem (1.4). This solution is positive and symmetric, has at most one critical pointin (0, 1), and satisfies

max

sinh(b1),−β≥ u′(x) ≥ −(α − αβ)

arsinh(−β)

αin (0, 1]

and√

1 + α2 − x2 ≥ u(x) ≥ Cα,β in [−1, 1],

with a constant Cα,β > 0 depending on Mα,β, α and −β.

37

Proof. By density of polynomials in H2(−1, 1) a minimising sequence (uk)k∈N for Mα,β may bechosen in Nα,β which consists of positive symmetric polynomials of degree at least two and suchthat Wh(uk) ≤ Mα,β +1 for all k ∈ N. By Proposition 4.38 we may assume that each element uk ofthe minimising sequence satisfies (4.18). The rest of the proof is along the lines of Theorem 3.11.Moreover, with the construction of Lemma 4.27 one can prove that u has at most one critical pointin (0, 1).

Reasoning as in the proof of Propositions 3.12 and 4.18, one can prove that the energy is strictlydecreasing in α.

Proposition 4.40. Let Mα,β be as defined in (2.7) and α∗ ≤ α < αβ. Then Meα,β > Mα,β for allα ∈ (α∗, α).

4.3.4 The case α < α∗

In this case no catenoid is going through the points (±1, α) which u ∈ Nα,β can be comparedwith. However, the results from the previous subsection will be useful also here. Since u ∈ Nα,β

is strictly positive, going from the right to the left, there exists certainly a first point x whereu(x) = α∗x. From here on, we may refer to the geometric constructions which led to suitableminimising sequences as described in Proposition 4.38.

The difficulty is now to understand how the graph of a suitable function u ∈ Nα,β shouldbehave or should be suitably modified before reaching the line y 7→ α∗y. By Lemma 4.34 it issufficient to consider functions where u′ 6= −β when we are below the line y 7→ αy. This resultgives a bound on the derivative when the graph of u is below the line y 7→ αy. Hence, it remains toget an estimate on u′ on the set x : αx < u(x) < α∗x. To this end we study the function u(x)/x.In order to ensure that u(x)/x has only finitely many oscillations in [0, 1] we restrict ourselves toadmissible functions as defined in Definition 4.26. Going from the left to the right, we prove thatthe function u(x)/x is decreasing from the first and only point where the graph of u crosses theline y 7→ α∗y to the point where it crosses or touches the line y 7→ αy. This leads to bounds forthe derivative on suitably modified minimising sequences.


We start by showing that, going from the right to the left, once the graph of u reaches the liney 7→ α∗y, then one may achieve that it remains above this line.

Lemma 4.41. Let u ∈ Nα,β be an admissible function in the sense of Definition 4.26 satisfying(4.17) for some x0 ∈ [0, 1). Let x be such that u(x) = α∗x and u(x) < α∗x in (x, 1].

Then there exists an admissible function v ∈ Nα,β with Wh(u) ≥ Wh(v) and satisfying (4.17) forsome x1 ∈ [0, 1). Moreover, v(x) = α∗x, v(x) < α∗x in (x, 1] and v(x) ≥ cosh(b∗x/x)x/b∗ > α∗xas well as v′(x) < α∗ in [0, x).

Proof. We have u′(x) ≤ α∗. If u′(x) < α∗ then u(x) > cosh(b∗x/x)x/b∗ in a left neighbourhood ofx. Since u satisfies (4.17) one sees as in the proof of Lemma 4.36 that u(x) > cosh(b∗x/x)x/b∗ >α∗x and u′(x) < α∗ for all x ∈ [0, x]. The claim then follows with v = u.

If instead u′(x) = α∗ then we substitute u in [−x, x] by cosh(b∗x/x)x/b∗. We get a newadmissible function v ∈ Nα,β with lower Willmore energy than u and v(x) > α∗x as well asv′(x) < α∗ in (0, x).

Notice that in the previous lemma if u(x) ≥ αx then also v(x) ≥ αx. Moreover, if u(x0) ≥ µx0

then also v(x1) ≥ µx1.

38

a)

α∗

α

u

0 1x b)

α∗

α

u

0 1x

cosh(λx)λ

with λ = b∗

x

Figure 11: Proof of Lemma 4.41 in the case where u ∈ Nα,β is tangent in x to the line y 7→ α∗y.

The aim of the next constructions is to show for functions u as in Lemma 4.41 one may achievethat u(x)/x is decreasing for x ∈ (0, 1) where αx < u(x) < α∗x. As in Lemma 4.32, they arebased on inserting parts of geodesic circles and catenoids to shorten the intervals and decreasethe Willmore energy. Assuming u(1) < α∗ we study now additional properties inherited by theshortened function u.

In the next result we prove that if u(x)/x is decreasing in x ∈ (0, 1) : u(1)x ≤ u(x) ≤ α∗xand u(x) ≥ u(1)x then, when ≥ u(1)/α∗, also u(x)/x is decreasing in x ∈ (0, ) : u(1)x ≤u(x) ≤ α∗x and u(x) ≥ u(1)x/. Notice that the result holds when we shorten the graph of uuntil we reach the line y 7→ α∗y but not until y 7→ αβy.

Proposition 4.42. Fix t < 0. Let u ∈ C1,1([−1, 1], (0,∞)) be an admissible function in the senseof Definition 4.26 satisfying (4.17) for some x0 ∈ [0, 1), u(x) ≥ u(1)x in [0, 1], u′(1) = −t andu(1) < α∗. Moreover, for ∈ [u(1)/α∗, 1], let u be the function constructed in Lemma 4.32. Then

u(x) ≥ u(1)

x for all x ∈ [0, ]

and u satisfies (4.17) on [0, ) for some x ∈ [0, ).Furthermore, if there exists x ∈ (0, 1) such that u(x) ≥ α∗x for all x ∈ [0, x], u(x) < α∗x for

all x ∈ (x, 1] and u(x)/x is decreasing in [x, 1], then there exists x ∈ (0, ) such that u(x) ≥ α∗xfor all x ∈ [0, x], u(x) < α∗x for all x ∈ (x, ] and u(x)/x is decreasing in [x, ].

Proof. For ∈ [u(1)/α∗, 1] there exists r′ ∈ [0, ) such that u is given in [0, ] by

u(x) =

u(1 + x − ) if r′ ≤ x ≤ ,f(x) if 0 ≤ x < r′,

where f is either an arc of a circle (and u′(r

′) ≤ 0) or a cosh(λx)/λ for λ ∈ R+. The first claim is

satisfied in [0, r′] since if f is a cosh then u(x) ≥ α∗x and α∗x ≥ u(1)x/ by assumption. On theother hand if f is a circular arc then

u(x)

x=

f(x)

x≥ f(r′)

r′=

u(r′ + 1 − )

r′≥ u(1)

r′ + 1 −

r′≥ u(1)

.

39

For x ∈ [r′, ] we have

u(x)

x=

u(x + 1 − )

x≥ u(1)

x + 1 −

x≥ u(1)

.

For the second claim let x be the biggest element in [0, ] such that u(x) ≥ α∗x in [0, x]. We firsttreat the case when f(x) = cosh(λx)/λ. Since cosh(λx)/λ ≥ α∗x then x ≥ r′. Moreover, sinceu(x + 1 − ) = u(x) = α∗x < α∗(x + 1 − ) we necessarily have x + 1 − > x. So, for x > x wehave

u(x) =u(x + 1 − )

x + 1 − (x + 1− ) ≤ u(x + 1 − )

x + 1 − (x + 1− ) =

α∗xx + 1 −

(x + 1− ) < α∗x, (4.20)

giving u(x) < α∗x in (x, ]. Since u(x)/x is decreasing in (x, 1] then u′(x) ≤ u(x)/x in (x, 1].Using that x < x + 1 − we find for x ∈ (x, ]

u′(x) = u′(x + 1 − ) ≤ u(x + 1 − )

x + 1 − =

u(x)

x

x

x + 1 − <

u(x)

x, (4.21)

showing that u(x)/x is decreasing in [x, ].If instead f(x) is a circular arc of a circle we need to distinguish two cases. If x ≥ r′ then we

reason as in (4.20) and (4.21). If instead x < r′ then u(x) < α∗x and u(x)/x is decreasing in(x, r′] since u′

≤ 0 in [0, r′]. In particular u(r′ + 1 − ) = u(r′) < α∗r′ and so r′ + 1 − > x. It

then follows for x ∈ [r′, ] that

u(x) =u(x + 1 − )

x + 1 − (x + 1 − ) ≤ u(r′ + 1 − )

r′ + 1 − (x + 1 − ) < α∗ r′(x + 1 − )

r′ + 1 − ≤ α∗x,

and proceeding as in (4.21) one shows that u(x)/x is decreasing also in [r′, ].

Thanks to the previous proposition we may now show that if u(x) ≥ u(1)x in [0, 1] andu(1) < α∗, then we can also assume that u(x)/x is decreasing on the set x ∈ (0, 1] : u(x) ≤ α∗x.

Proposition 4.43. Let u ∈ C1,1([−1, 1], (0,∞)) be a admissible function in the sense of Definition4.26 satisfying (4.17) for some x0 ∈ [0, 1), u′(1) > 0 and u(1) < α∗. We assume further thatu(x) > u(1)x in (0, 1) and that there exists x ∈ (0, 1) with u(x) ≥ α∗x in [0, x] and u(x) < α∗x in(x, 1].

Then, there exists a admissible function v ∈ C1,1([−1, 1], (0,∞)) with v′(1) = u′(1), u(1) =v(1), Wh(u) ≥ Wh(v) and satisfying (4.17) for some x ∈ [0, 1). Moreover, there exists x ∈ (0, 1)so that v(x) ≥ α∗x in [0, x] and v(x) < α∗x for all x ∈ (x, 1] and v(x)/x is decreasing in [x, 1].

Proof. By assumption u(x)/x < u(x)/x in a right neighbourhood of x and u(x)/x > u(1)/1 in aleft neighbourhood of 1. If u(x)/x is decreasing in [x, 1] the claim follows with v = u. Otherwisethere exists a first local minimum x1 of u(x)/x in [x, 1]. By our definition of admissibility thisminimum is strict. For easy notation let α′ denote u(x1)/x1. Notice that u(1) < α′ < α∗ and thatu′(x1) = α′. Let x3 be the smallest element in (x1, 1] with u(x3) = α′x3 and x2 be the largestelement in (x1, x3) with u′(x2) = α′. Then u(x) > α′x for x ∈ (x1, x3). For x ∈ (x2, x3) we seethat x2(u

x)′ = xu′ − u < xα′ − xα′ = 0, i.e. x 7→ ux is strictly decreasing on (x2, x3) so that it has

a local maximum on (x1, x2).The idea is to replace u|[−x2,x2] by the appropriately shortened and rescaled u|[−x1,x1] according

to Proposition 4.42. For this new function v the number of local extrema of x 7→ v(x)/x belowthe line y 7→ α∗y has decreased by at least two. Since x 7→ u(x)/x has only finitely many local

40

a)

α∗

α′

u(1)

0 1x1 x2 x3 b)

α∗

α′

u(1)

0 1x1 x2 x3

Figure 12: Proof of Proposition 4.43. In x1 and x2 the tangents are parallel. u|[−x1,x1] is shortenedand rescaled. So, we avoid oscillations of x 7→ u(x)/x decreasing the Willmore energy.

extrema, after finitely many iterations we obtain the claim. We present this argument now indetail.

By using a scaled version of Proposition 4.42 we shorten the function u|[−x1,x1] by insertinga cosh or an arc of a circle. For each ∈ [u(x1)/α∗, x1] there exists a symmetric admissiblew ∈ C1,1([−, ], (0,∞)) with w(±) = u(x1), w′

() = α′ = u′(x1) and lower Willmore energythan u|[−x1,x1]. We choose = u(x1)x2/u(x2) so that we shorten the graph of u|[−x1,x1] until

the line y 7→ u(x2)y/x2 is reached. By u(x2)x2

> u(x1)x1

and u(x2) < α∗x2 we see that indeed ∈ [u(x1)/α∗, x1]. Moreover, since u(x) ≥ α∗x in [0, x] and u(x)/x is decreasing in [x, x1] thenby Proposition 4.42, there exists x′ such that w(x) ≥ α∗x in [0, x′] and w(x)/x is decreasing in[x′, ]. The function v equal to u in [x2, 1] and to the rescaled w in [0, x2] is admissible and hasthe same boundary values as u. Finally, v satisfies (4.17) for some x ∈ [0, 1) and there exists xsuch that v(x) ≥ α∗x in [0, x] and v(x) < α∗x in [x, 1] and v(x)/x has on [x, 1] at least two localextrema less than u(x)/x in [x, 1].

Since u(x)/x has only finitely many local extrema, the claim is proved by finitely many itera-tions.

Notice that if u(x0) ≥ µx0 then also v(x) ≥ µx.The previous proposition is the main ingredient which allows us to pass to functions with

uniformly bounded derivatives. For this purpose, we distinguish again the cases α ≥ −β andα < −β.

The case α ≥ −β

In the next lemma we prove that in the case α ≥ −β it is sufficient to consider functions satisfyingu(x) ≥ αx in [0, 1].

Lemma 4.44. We assume in addition that α ≥ −β. Let u ∈ Nα,β be an admissible function inthe sense of Definition 4.26 satisfying (4.17) for some x0 ∈ [0, 1) and u′(x) 6= −β for all x ∈ [0, 1)with u(x) ≤ αx.

Then, there exists an admissible function v ∈ Nα,β with lower Willmore energy than u, satis-fying v(x) > αx for all x ∈ [0, 1) and (4.17) for some x′ ∈ [0, 1).

Proof. We assume first that even α > −β so that u(x) > αx in a left neighbourhood of 1. Ifu(x) ≤ αx for some x ∈ (0, 1) then there exists a smallest element x1 in [0, 1] such that u(x1) = αx1.Let x2 ≥ x1 be the smallest element such that u′(x2) = α and u(x2) ≤ αx2. If u(x)/x ≥ u(x2)/x2

41

for all x ∈ (0, x2] we denote x2 by x. Otherwise, x denotes the largest element in (0, x2) such thatu(x)/x ≥ u(x)/x for all x ∈ (0, x]. Then u(x) ≤ αx, u′(x) ≤ α and, by assumption, u′(x) > −β.Let x ∈ [x2, 1] be the biggest element such that u′(x) = u′(x) and u(x) > αx. Such an elementexists since −β < α.

We notice first that u(x) < α∗x. Indeed, if u(x) ≥ α∗x since u′(x) ≤ α < α∗ then u(x) > α∗xin a left neighbourhood of x and by Lemma 4.30 for these points u′(x) < α∗ and by continuity ofu and of its derivative then u(x) ≥ α∗x for all x ∈ [0, x]. This contradicts the assumption thatu(x) < αx in some interval.

The construction is now done similarly to Proposition 4.43. By using a scaled version ofProposition 4.42 we shorten the function u|[−x,x] by inserting a cosh or an arc of a circle. Weshorten it until we reach the line y 7→ u(x)y/x. That is, we consider the function w with = u(x)x/u(x) constructed by a rescaled version of Proposition 4.42 applied to u|[−x,x]. Sinceu(x)/x ≥ u(x)/x in (0, x] then by the first claim in Proposition 4.42, we have w(x) ≥ w()x/in (0, ]. Hence, the function v which is equal to u in [x, 1] and equal to the rescaled w in [0, x]yields the claim.

If α = −β, let x ∈ (0, 1] be the smallest element such that v(x) = αx. By the assumptions itfollows that x = 1. Indeed, if x < 1 then, u′(x) ≤ α = −β and the assumption gives u′(x) < −β =α. But there exists then x′ > x such that v′(x′) = α = −β and v(x′) < αx′, a contradiction.

We may now assume that u(x) ≥ αx in [0, 1]. In the next corollary, we first observe that theset x ∈ [0, 1] : αx ≤ u(x) ≤ α∗x is an interval and then, by using Proposition 4.43, we show thatwe may assume that in this interval u(x)/x is decreasing. This yields suitable a priori bounds.

Corollary 4.45. Let α be such that α ≥ −β. Let u ∈ Nα,β be an admissible function in thesense of Definition 4.26 satisfying (4.17) for some x0 ∈ [0, 1), u′(x) 6= −β for all x ∈ [0, 1) withu(x) ≤ αx.

Then, there exists an admissible function v ∈ Nα,β with lower Willmore energy than u andsatisfying v′(x) ≤ α∗ in [0, 1] and v(x) > αx in [0, 1) as well as (4.17) for some x1 ∈ [0, 1).

Proof. By Lemmas 4.44 and 4.41 and the following remark, there exists w ∈ Nα,β with lowerWillmore energy than u such that w(x) > αx in [0, 1), w satisfies (4.17) for some x′ ∈ [0, 1) and sothat there exists x ∈ (0, 1) such that w(x) ≥ α∗x in [0, x] and w(x) < α∗x in (x, 1]. By Proposition4.43 there exists v ∈ Nα,β with lower Willmore energy than w such that v satisfies (4.17) forsome x1 ∈ [0, 1) and so that there exists x2 ∈ (0, 1) such that v(x) ≥ α∗x in [0, x2] and v(x)/x isdecreasing in [x2, 1]. This shows in particular that v(x) > αx in [0, 1). Since v′(x) ≤ v(x)/x in[x2, 1] we find v′(x) ≤ α∗ in [x2, 1]. Reasoning as in Lemma 4.41 we get v′(x) ≤ α∗ in [0, 1].

The case α < −β

The next result corresponds to Corollary 4.45. We first observe that we have a bound on thederivative when the graph of u is below the line y 7→ αy. Then, when the graph of u crosses thisline, we are back in the previous case.

Corollary 4.46. We assume in addition that −β > α. Let u ∈ Nα,β be an admissible function inthe sense of Definition 4.26 satisfying (4.17) for some x0 ∈ [0, 1) and u′(x) 6= −β for all x ∈ [0, 1)with u(x) ≤ αx.

Then, there exists an admissible function v ∈ Nα,β with lower Willmore energy than u, satis-fying (4.17) for some x1 ∈ [0, 1) and v′(x) ≤ max−β, α∗ on [0, 1].

Proof. Let w ∈ Nα,β be the function constructed in Lemma 4.41 such that Wh(w) ≤ Wh(u),w satisfies (4.17) for some x ∈ [0, 1) and there exists x ∈ (0, 1) with w(x) ≥ α∗x in [0, x] and

42

w(x) < α∗x in (x, 1]. Notice that by the construction also w satisfies that w′(x) 6= −β for allx ∈ [0, 1) with w(x) < αx. Let x2 be the biggest element in (0, 1) such that w(x2) = αx2. Thenfor all x ∈ (x2, 1) we have w(x) < αx and w′(x) < −β.

If x ≥ x2 the claim follows with v = w and x1 = x. Otherwise, since w(x2) = αx2 < α∗x2

and w′(x2) ≤ α then w(x2)/x2 ≥ w′(x2). Hence, applying a rescaled version of Lemma 4.44 andof Corollary 4.45 to w|[−x2,x2] we find an admissible function v ∈ C1,1([−x2, x2], (0,∞)) with lowerWillmore energy than w|[−x2,x2] with the same boundary values and such that v(x) > αx in [0, x2)and v′(x) ≤ α∗ in [0, x2). Defining v(x) = w(x) for x ∈ [x2, 1] and extending v by symmetry tothe interval [−1, 1] yields the claim.

Characterisation of suitable minimising sequences

The following proposition characterises suitably modified minimising sequences. We do not needto distinguish the cases α ≥ −β and α < −β.

Proposition 4.47. Let (uk)k∈N be a minimising sequence for Mα,β in Nα,β of admissible functionsin the sense of Definition 4.26 such that Wh(uk) ≤ Mα,β + 1 for all k ∈ N. Then, there existsa minimising sequence (vk)k∈N ⊂ Nα,β of admissible functions having lower Willmore energy andsatisfying (4.17) as well as

max−β, α∗ ≥ v′k(x) ≥ −arsinh(−β)

α(αβ − α) and Cα,β ≤ vk(x) ≤

√1 + α2 − x2, (4.22)

in [0, 1] with a constant Cα,β > 0 depending on α, −β and Mα,β.

Proof. By Lemmas 4.28, 4.29, 4.34, 4.44 and Corollary 4.45 if −β ≤ α or Corollary 4.46 if −β > α,for each uk there exists an admissible vk ∈ Nα,β with lower Willmore energy than uk satisfying(4.17) for some xk ∈ [0, 1) and

v′k(x) ≤ max−β, α∗ and vk(xk) ≥α

arsinh(−β)(αβ − α)xk. (4.23)

Since vk satisfies (4.16) we get vk(x) ≤√

1 + α2 − x2 for x ∈ [0, 1] and

v′k(x) ≥ − x

vk(x)≥ − xk

vk(xk)≥ −arsinh(−β)

αβ − α

αfor x ∈ [0, xk],

while v′k ≥ 0 in [xk, 1]. The estimate from below for vk follows from the second estimate in (4.23)if xk ≥ 1/2 and from Lemma 4.9 if xk ≤ 1/2.


Theorem 4.48 (Existence and regularity). For β < 0 and α < α∗ there exists a symmetricfunction u ∈ C∞([−1, 1], (0,∞)) such that the corresponding surface of revolution Γ ⊂ R

3 solvesthe Dirichlet problem (1.4). This solution u has at most one critical point in (0, 1) and obeys thefollowing estimates:

maxα∗,−β

≥ u′(x) ≥ −(α − αβ)

arsinh(−β)

αin (0, 1]

and√

1 + α2 − x2 ≥ u(x) ≥ Cα,β in [−1, 1],

with a constant Cα,β > 0 depending on Mα,β, α and −β.

43

Proof. By density of polynomials in H2(−1, 1) a minimising sequence (uk)k∈N for Mα,β may bechosen in Nα,β which consists of positive symmetric polynomials of degree at least 2 and obeysWh(uk) ≤ Mα,β + 1 for all k ∈ N. By Proposition 4.47, there exists a minimising sequence(vk)k∈N ⊂ Nα,β such that Wh(vk) ≤ Wh(uk) and each element vk of the minimising sequencesatisfies (4.22). The rest of the proof is along the lines of Theorem 3.11. Moreover, Lemma 3.20shows that u has at most one critical point in (0, 1).

Reasoning as in the proof of Propositions 3.12 and 4.18, one can prove that also in this casethe energy is strictly decreasing in α.

Proposition 4.49. Let Mα,β be as defined in (2.7) and α < α∗. Then Meα,β > Mα,β for all α < α.

5 Convergence to the sphere for α ց 0

In this section, we choose any β ∈ R, keep it fixed and study the singular limit α ց 0, where the“holes” ±1 × Bα(0) in the cylindrical surfaces of revolution disappear.

The aim of this chapter is to show that if uα ∈ Nα,β is an energy minimising solution to (1.4),i.e. Wh(uα) = Mα,β , then uα converges for α ց 0 to the semicircle

√1 − x2. So, the surface

of revolution generated by the graph of uα converges to the sphere, which shows up as a limitirrespective of the prescribed boundary slope ±β.

We first show that for α small, any minimiser uα ∈ Nα,β of Wh, i.e. Wh(uα) = Mα,β , has thesame qualitative properties as the solution we have constructed.

Lemma 5.1. We assume that α < minα∗, 1/β if β > 0 and α < α∗ if β ≤ 0. Let u ∈ Nα,β

be such that Wh(u) = Mα,β. Then, u ∈ C∞([−1, 1], (0,∞)) and u has the following additionalproperties:

1. If β ≥ 0, then u′ < 0 in (0, 1) and

α ≤ u(x) ≤√

1 + α2 − x2 in [−1, 1], x + u(x)u′(x) > 0 in (0, 1).

2. If β < 0, then u has at most one critical point in (0, 1), i.e. there exists x0 ∈ [0, 1) such thatu′ > 0 in (x0, 1], u′(x0) = 0 and u′ < 0 in (0, x0). Moreover,

x + u(x)u′(x) > 0 in (0, 1], u′(x) ≤ γ := max−β, α∗ in [x0, 1]

and u(x) ≥ min

1

2

α

arsinh(−β)(αβ − α),1

2

γ

eC − 1

in [−1, 1],

with C =γ√

1+γ2

2

(Mα,β + 4β√

1+β2

)> 0.

Proof. Since u minimises Wh and hence, weakly solves the Euler-Lagrange equation (2.5), theargument in [DDG, Theorem 3.9, Step 2] yields u ∈ C∞([−1, 1], (0,∞)). Whenever x0 ∈ (0, 1) is acritical point of u one may insert on [−x0, x0] a rescaled energy minimising solution v according toTheorem 3.18 and Lemma 3.20 satisfying v′ < 0 on (0, x0). Putting together v and u|[−1,1]\[−x0,x0]

yields a further minimiser in Nα,β and so, a solution to (2.5). By uniqueness of the initial valueproblem, this new solution coincides with the original u. This shows that u has at most one criticalpoint. Exploiting this observation, one proves that u satisfies the estimate in the claim by the sameconstructions as in the proof of the respective existence theorems.

44

The assumption on α excludes in particular the case 0 < −β < α and α > αβ , where “our”solution was constructed in a much smaller set than Nα,β .

In what follows, when considering minimisers u of Wh in Nα,β , we make use of the qualitativeproperties of u stated in Lemma 5.1 without further notice. In particular, we restrict ourselvesalways to the case α < minα∗, 1/|β|.

5.1 An upper bound for the energy as α ց 0

For α small we first construct a function fα ∈ Nα,β such that its Willmore energy converges to theone of the sphere for α ց 0. We consider the symmetric function:

fα(x) =

α√1+β2

cosh

(√1+β2

α (x − x1)

)if x0 ≤ x ≤ 1,

√r2 − x2 if − x0 < x < x0,

α√1+β2

cosh

(√1+β2

α (x + x1)

)if − 1 ≤ x ≤ −x0,

(5.1)

where x1 = 1 − α arsinh(−β)/√

1 + β2, r2 = x20 + fα(x0)

2 and, for α small enough, x0 ∈ (0, 1) issolution of

−x0 =α

2√

1 + β2sinh

(2

√1 + β2

α(x0 − x1)

). (5.2)

Assuming α to be small enough ensures the existence of x0 ∈ (0, 1). We remark that this x0 shouldnot be mixed with the one in Condition (4.17). The function fα has Willmore energy

Wh(fα) = 2

1∫

x0

κh[fα]2√

1 + f ′2α (x)

fα(x)dx = 8

1∫

x0

1

cosh2(

√1+β2

α (x − x1))

√1 + β2

αdx

= 8 tanh(arsinh(−β)) − 8 tanh

(√1 + β2(x0 − x1)

α

)

= −8β√

1 + β2+ 8 tanh

(√1 + β2(x1 − x0)

α

)≤ −8

β√1 + β2

+ 8.

This particular function shows that Mα,β is uniformly bounded for α going to 0 . Since Mα,β isincreasing for α ց 0 for all β ∈ R, it follows that

limαց0

Mα,β ≤ 8 − 8β√

1 + β2. (5.3)

5.2 The limit of the energy

In this section we prove that the limit of the energy is equal to the upper bound given in theprevious section.

We start by proving that when β < 0 and α is small the minimiser has precisely one criticalpoint in (0, 1) and this point approaches 1 for α going to 0.

Lemma 5.2. Let β < 0. We assume that uα ∈ Nα,β minimises the Willmore energy, i.e.Wh(uα) = Mα,β. Let xα ∈ [0, 1) be such that u′

α(xα) = 0 and u′α > 0 in (xα, 1]. Then,

limαց0

xα = 1.

45

Proof. Let us assume that there exist a sequence αk ց 0 and δ > 0 such that xαk≤ 1 − δ for all

k ∈ N. For each α > 0, the measure of the set A := x ∈ [1 − δ, 1] : u′α(x) ≥ 2α/δ is bounded by

δ/2, since uα is strictly positive. Then looking at the energy we find

Mαk,β = Wh(uαk) ≥

1∫

xαk

1

uαk(x)

√1 + u′2

αk(x)

dx + 2u′

αk(x)√

1 + u′αk

(x)2

∣∣∣1

−1

≥∫

[1−δ,1]\A

1

αk

√1 + u′

αk(x)2

dx − 4β√

1 + β2

≥ δ

2αk

√1 +

4α2k

δ2

− 4β√

1 + β2→ ∞ for k → ∞,

a contradiction to (5.3).

We show that the gradient of any minimiser is unbounded near x = −1 in the limit α ց 0.From now on, β is a fixed element of R.

Lemma 5.3. We fix δ0 ∈ (0, 1). For α > 0, let uα ∈ Nα,β be a minimiser of the Willmore energy,i.e. Wh(uα) = Mα,β. Then,

limαց0

maxx∈[−1,−1+δ0]

u′α(x) = +∞.

Proof. We assume by contradiction that there exist a sequence αk ց 0 and a positive constant Ksuch that

maxx∈[−1,−1+δ0]

u′αk

(x) ≤ K for all k ∈ N. (5.4)

Let xαk= 1 if β ≥ 0. If β < 0, let xαk

∈ [0, 1) be the element such that u′αk

(xαk) = 0 and u′

αk> 0

in (xαk, 1]. By Lemma 5.2 we have 1− xαk

→ 0 for k → ∞. Notice that uαk(xαk

) ≤ αk. Then, weestimate the Willmore energy from below as follows

Wh(uαk) ≥

1∫

−1

1

uαk(x)(1 + u′

αk(x)2)

1

2

dx − 4β√

1 + β2

≥ 2

−1+δ0∫

−xαk

1

uαk(x)(1 + u′

αk(x)2)

1

2

dx − 4β√

1 + β2. (5.5)

By (5.4) we have uαk(x) ≤ uαk

(xαk) + K(x + 1) for x ∈ [−xαk

,−1 + δ0] and hence from (5.5) weconclude for k large enough

Wh(uαk) ≥ 2√

1 + K2

−1+δ0∫

−xαk

1

uαk(xαk

) + K(x + 1)dx − 4

β√1 + β2

=2

K√

1 + K2log

(1 + K

δ0 − 1 + xαk

uαk(xαk

) + K(1 − xαk)

)− 4

β√1 + β2

≥ 2

K√

1 + K2log

(1 +

K

2

δ0

uαk(xαk

) + K(1 − xαk)

)− 4

β√1 + β2

.

Since uαk(xαk

) ≤ αk ց 0 and xαk→ 1 for k → ∞, the energy Wh(uαk

) diverges to +∞, therebycontradicting estimate (5.3).

46

Theorem 5.4 (Limit of the energy for α ց 0). For α > 0 let uα ∈ Nα,β be such that Wh(uα) =Mα,β. Then, it holds that

limαց0

Wh(uα) = limαց0

Mα,β = 8 − 8β√1 + β2

.

Proof. For any u ∈ Nα,β and δ0 > 0 to be chosen we have

1

2Wh(u) =

−1+δ0∫

−1

(u′′(x)

(1 + u′(x)2)3

2

− 1

u(x)(1 + u′(x)2)1

2

)2

u(x)√

1 + u′(x)2 dx

+ 4

−1+δ0∫

−1

u′′(x)

(1 + u′(x)2)3

2

dx +

0∫

−1+δ0

κh[u]2√

1 + u′(x)2

u(x)dx

≥ 4

−1+δ0∫

−1

u′′(x)

(1 + u′(x)2)3

2

dx = 4u′(x)√

1 + u′(x)2

∣∣∣−1+δ0

−1

= 4u′(−1 + δ0)√

1 + u′(−1 + δ0)2− 4

β√1 + β2

. (5.6)

Let αk ց 0 be any sequence. By Lemma 5.3 we find δαk∈ [0, 1/2] with limk→∞ u′

αk(−1+δαk

) = ∞.From (5.6) it follows with δ0 = δαk

Wh(uαk) ≥ 8

u′αk

(−1 + δαk)√

1 + u′αk

(−1 + δαk)2

− 8β√

1 + β2,

and hence

limk→∞

Wh(uαk) ≥ 8 − 8

β√1 + β2

.

This estimate together with (5.3) yields the claim.

Corollary 5.5. For α > 0 let uα ∈ Nα,β be such that Wh(uα) = Mα,β. Then,

limαց0

1−δ0∫

−1+δ0

κh[uα]2√

1 + u′α(x)2

uα(x)dx = 0 for all δ0 ∈ (0, 1).

Proof. For any sequence αk ց 0, by Lemma 5.3 there exist δαk∈ [0, δ0] with

limk→∞

u′αk

(−1 + δαk) = +∞. (5.7)

Proceeding similarly as in the proof of Theorem 5.4 we have

Wh(uαk) ≥

1−δαk∫

−1+δαk

κh[uαk]2

√1 + u′

αk(x)2

uαk(x)

dx + 8u′

αk(−1 + δαk

)√1 + u′

αk(−1 + δαk

)2− 8

β√1 + β2

.

Since

0 ≤1−δ0∫

−1+δ0

κh[uαk]2

√1 + u′

αk(x)2

uαk(x)

dx ≤1−δαk∫

−1+δαk

κh[uαk]2

√1 + u′

αk(x)2

uαk(x)

dx

the claim follows from the inequalities above, Theorem 5.4 and (5.7).

47

5.3 The minimiser converges to the sphere

Lemma 5.6. Fix δ0 ∈ (0, 1). For α > 0 let uα ∈ Nα,β solve Wh(uα) = Mα,β. Then, there existsε > 0 such that uα(x) ≥ ε in [−1 + δ0, 1 − δ0] for all α ≤ 1.

Proof. We assume by contradiction that there is a sequence 1 ≥ αk ց 0 and that there are pointsxk ∈ [0, 1 − δ0] with 1 ≥ uαk

(xk) = minx∈[0,1−δ0] uαk(x) =: mk ց 0. The energy of this sequence

of minimisers is bounded from below as follows

Wh(uαk) ≥

1∫

−1

1

uαk(x)

√1 + u′

αk(x)2

dx − 4β√

1 + β2

≥ 2

1∫

1−δ0

1

uαk(x)

√1 + u′

αk(x)2

dx − 4β√

1 + β2

≥ 2

maxmk, αk

1∫

1−δ0

1√1 + u′

αk(x)2

dx − 4β√

1 + β2. (5.8)

In order to estimate the integral in (5.8), we apply the Cauchy-Schwarz inequality

δ0 =

1∫

1−δ0

1

(1 + u′αk

(x)2)1

4

(1 + u′αk

(x)2)1

4 dx

≤

1∫

1−δ0

1√1 + u′

αk(x)2

dx

1

2

1∫

1−δ0

√1 + u′

αk(x)2 dx

1

2

,

which implies

δ20 ≤

δ0 +

1∫

1−δ0

|u′αk

(x)| dx

1∫

1−δ0

1√1 + u′

αk(x)2

dx. (5.9)

We estimate the first integral. Let xαk= 1 if β ≥ 0 and if β < 0 let xαk

∈ [0, 1) be the elementsuch that u′

αk(xαk

) = 0 and u′αk

> 0 in (xαk, 1]. Lemma 5.2 shows that xαk

≥ 1−δ0 for sufficientlylarge k. Splitting the integral we find

1∫

1−δ0

|u′αk

(x)| dx ≤xαk∫

xk

|u′αk

(x)| dx +

1∫

xαk

|u′αk

(x)| dx =

mk − αk if β ≥ 0,mk − 2uαk

(xαk) + αk if β < 0.

Estimating the right hand side in the inequality above by mk + αk, we then conclude from (5.9)that

δ20 ≤ (δ0 + mk + αk)

1∫

1−δ0

1√1 + u′

αk(x)2

dx.

Inserting this into (5.8) yields for k sufficiently large

Wh(uαk) ≥ 2δ2

0

maxmk, αk (δ0 + mk + αk)− 4

β√1 + β2

≥ 2δ20

maxmk, αk (δ0 + 2)− 4

β√1 + β2

−→ ∞ for k → ∞,

48

contradicting Theorem 5.4.

Corollary 5.7. Fix δ0 ∈ (0, 1). For α > 0 small enough let uα ∈ Nα,β solve Wh(uα) = Mα,β.Then, there exists ε > 0 such that

−1

ε≤ u′

α(x) ≤ maxα∗,−β for all x ∈ [0, 1 − δ0],

where α∗ = mincosh(b)/b : b > 0.

Proof. The first inequality is a consequence of Lemma 5.6 and the inequality 0 ≤ x + u(x)u′(x) in[0, 1]. The second one follows from the estimates on the minimiser in Lemma 5.1.

Theorem 5.8 (Convergence to the sphere). For α > 0 sufficiently small let uα ∈ Nα,β be aminimiser of the Willmore energy, i.e. Wh(uα) = Mα,β. Let u0 denote the semicircle u0(x) :=√

1 − x2, x ∈ [−1, 1]. Then, for any m ∈ N,

limαց0

uα = u0 in Cmloc(−1, 1).

Proof. We choose any δ0 ∈ (0, 1). Let (αk)k∈N be any sequence with αk ց 0. By Lemma 5.6 andCorollary 5.7 there exists a ε > 0 such that

ε ≤ uαk(x) ≤

√1 + α2

k − x2 and − 1

ε≤ u′

αk(x) ≤ 1

ε,

for x ∈ [−1 + δ0, 1 − δ0] and k sufficiently large. By these uniform bounds, the monotonicity in αof the energy, and Theorem 5.4 we find

8 − 8β√

1 + β2≥ Wh(uαk

)

≥1−δ0∫

−1+δ0

u′′αk

(x)2uαk(x)

(1 + u′αk

(x)2)5

2

dx +

1−δ0∫

−1+δ0

1

uαk(x)

√1 + u′

αk(x)2

dx − 4β√

1 + β2

≥ ε

(1 + 1ε2 )

5

2

1−δ0∫

−1+δ0

u′′2αk

dx +2√

1 + α2k

1

(1 + 1ε2 )

1

2

− 4β√

1 + β2.

Hence, (uαk)k∈N is uniformly bounded in H2(−1 + δ0, 1 − δ0). So, there exists a subsequence

(αkj)j∈N and a function u0 ∈ H2(−1 + δ0, 1 − δ0) such that

u′′αkj

u′′0 in L2(−1 + δ0, 1 − δ0) and uαkj

→ u0 in C1([−1 + δ0, 1 − δ0], (0,∞)).

Moreover, u0 satisfies: ε ≤ u0(x) ≤√

1 − x2, |u′0(x)| ≤ 1

ε for all x ∈ [−1 + δ0, 1 − δ0] and byCorollary 5.5

0 = lim infj→∞

1−δ0∫

−1+δ0

κh[uαkj]2

√1 + u′

αkj(x)2

uαkj(x)

dx ≥1−δ0∫

−1+δ0

κh[u0]2

√1 + u′

0(x)2

u0(x)dx.

Hence, κh[u0] ≡ 0 on [−1 + δ0, 1 − δ0] and, therefore, u0|[−1+δ0,1−δ0] is an arc of a geodesic circle,

i.e., since u0 is also symmetric around 0, it exists a radius r > 0 such that u0(x) =√

r2 − x2 in

49

[−1+ δ0, 1− δ0]. Necessarily r ≥ 1− δ0 and by the arbitrariness of δ0 we have r ≥ 1. On the otherhand, since u0(x) ≤

√1 − x2 we have r ≤ 1. Hence, r = 1 and u0 = u0.

Since for any sequence (αk)k∈N there exists a subsequence (αkj)j∈N such that uαkj

converges

to u0, we have that also uαkconverges to u0. The sequence being arbitrary, convergence in

C1 ∩H2([−1 + δ0, 1− δ0]) follows. Proceeding as in the proof of regularity in [DDG, Theorem 3.9]we conclude from the weak form of the differential equation (2.5) that uα → u0 for α ց 0 also inCm

loc(−1, 1) for any m ∈ N.

6 Qualitative properties of minimisers and estimates of the en-

ergy

In this section we give upper and lower bounds of the energy and we study the sign of the hyper-bolic curvature of our minimiser. We have to distinguish the cases as in the proof of existence.We remark that any minimiser in the respective class of admissible functions has the qualitativeproperties mentioned in the existence theorems, since our geometric constructions apply also tothese minimisers.

6.1 The case αβ > 1

6.1.1 Bounds on the energy

Proposition 6.1 (Upper bound of the energy). We have

Mα,β ≤ 2(αβ − 1)√1 + β2

arcsinβ√

1 + β2.

Proof. We consider the arc w ∈ Nα,β of the circle with centre in (0, α − 1/β) and radius√

1 + 1β2

which is given by

w(x) := α − 1

β+

√1 +

1

β2− x2 for x ∈ [−1, 1]. (6.1)

Its hyperbolic curvature is

κh[w](x) = − αβ − 1√1 + β2

for all x ∈ [−1, 1].

The previous identity for κh[w] implies

Wh(w) =αβ − 1√1 + β2

1∫

−1

1√

1 + 1β2 − x2

− 1

α − 1β +

√1 + 1

β2 − x2

dx

≤ αβ − 1√1 + β2

1∫

−1

1√1 + 1

β2 − x2dx.

The claim follows computing the integral.

The previous proposition indicates that possibly limα→∞ Mα,β = ∞. To prove this, we establisha lower bound for Mα,β for large α. In what follows G denotes the function G : R → (−c0/2, c0/2),

G(t) =

t∫

0

1

(1 + τ2)5

4

dτ with c0 :=

∫

R

1

(1 + τ2)5

4

dτ = B(1/2, 3/4

)= 2.39628 . . . , (6.2)

50

with B( . , . ) the Beta-Function. The G-function played an important role in the study of theone-dimensional Willmore problem, i.e. of so called elastica, see [E, pp. 233–234] and [DG1, DG2].

The following estimate from below holds true for any β 6= 0, irrespective of its sign.

Proposition 6.2 (A lower bound for the energy). Let β ∈ R \ 0. For α > 2|β| it holds that

Mα,β ≥ α min

G(−β)2, (G(α/2) + G(β))2− 4

β√1 + β2

,

where the function G is defined in (6.2). In particular, limα→∞ Mα,β = ∞.

Proof. For any u ∈ Nα,β we have

Wh(u) ≥1∫

−1

u′′(x)2u(x)

(1 + u′(x)2)5

2

dx − 4β√

1 + β2.

If u(x) ≥ α/2 for all x ∈ [−1, 1] then

Wh(u) ≥ α

2

1∫

−1

u′′(x)2

(1 + u′(x)2)5

2

dx − 4β√

1 + β2

≥ α

4

1∫

−1

u′′(x)

(1 + u′(x)2)5

4

dx

2

− 4β√

1 + β2

= α(G(−β))2 − 4β√

1 + β2.

Otherwise, there exists x1 ∈ (0, 1) such that u(x) ≥ α/2 in [x1, 1] and u′(x1) ≥ α/2. Proceedingsimilarly as before, Cauchy’s inequality yields

Wh(u) ≥ 2α

2

1∫

x1

u′′(x)2

(1 + u′(x)2)5

2

dx − 4β√

1 + β2

≥ α

1 − x1

1∫

x1

u′′(x)

(1 + u′(x)2)5

4

dx

2

− 4β√

1 + β2

≥ α(G(α/2) + G(β))2 − 4β√

1 + β2.

One should observe that the function G is strictly increasing. In both cases we have that

Wh(u) ≥ α min

G(β)2, (G(α/2) + G(β))2− 4

β√1 + β2

.

According to Propositions 6.1 and 6.2, Mα,β grows linearly in α → ∞. We recall that for β = 0the situation is different since Mα,0 → 0, see [DDG, Lemma 3.2] and also Proposition 6.6 below.

51

6.1.2 On the sign of the hyperbolic curvature of minimisers

From Proposition 3.7 we infer the following convexity property for minimisers of the Willmorefunctional.

Lemma 6.3. Let u ∈ Nα,β be a minimiser for Mα,β. Then, x 7→ u′(x) is strictly decreasing on[0, 1].

Proof. Since each minimiser satisfies x + u(x)u′(x) < 0 in (0, 1], the function u′ is decreasing in aright neighbourhood of 0. We assume by contradiction that u′ is not strictly decreasing on (0, 1].Then, there exist 0 < x1 < x2 ≤ 1 such that u′(x1) = u′(x2). We consider u|[−x1,x1] and rescale itto a function w ∈ C1,1([−x2, x2], (0,∞)). This function satisfies w(x2) = x2

x1u(x1) > u(x1) ≥ u(x2)

and w′(x2) = u′(x2). Moreover, by Remark 2.4 we conclude that

x2∫

−x2

κh[w]2 dsh[w] =

x1∫

−x1

κh[u]2 dsh[u] <

x2∫

−x2

κh[u]2 dsh[u], (6.3)

since κh[u] is not identically zero in [x1, x2]. This follows since u solves the differential equation(2.5) and is not part of a geodesic circle. On the other hand, exploiting that u is a minimiser, wefind

x2∫

−x2

κh[u]2 dsh[u] = inf

x2∫

−x2

κh[v]2 dsh[v] : v ∈ C1,1([−x2, x2], (0,∞)), symmetric,

v(x2) = u(x2) and v′(x2) = u′(x2)

≤x2∫

−x2

κh[w]2 dsh[w],

where in the last step we used that w(x2) > u(x2) and Proposition 3.7. One should observe thatthe condition 0 > x + u(x)u′(x) is the rescaled version of αβ > 1 on the interval [0, x]. We haveachieved a contradiction to (6.3).

Theorem 6.4. We assume that αβ > 1. Let u ∈ Nα,β be a minimiser for Mα,β. Then, eitherκh[u] < 0 in [0, 1), or there exists a ∈ (0, 1) such that κh[u] < 0 in [0, a) and κh[u] > 0 in (a, 1).

Proof. We consider the auxiliary function ϕ : [0, 1] → R defined by ϕ(x) := x + u(x)u′(x), where(ϕ(x), 0) is the centre and r(x) := u(x)

√1 + u′(x)2, x ∈ [0, 1] the radius of the geodesic circle being

tangential to the graph of u in (x, u(x)). From (3.2) we know that ϕ(x) < 0 in (0, 1], and ϕ(0) = 0.

Hence, ϕ(x) is decreasing for x > 0 sufficiently small. Since ϕ′(x) =1 + u′(x)2

3

2 κh[u](x), itfollows that κh[u] < 0 in a right neighbourhood of 0. Viewing at (2.5) now as a second orderequation for κh[u] satisfying a strong maximum / minimum principle provided that maxima /minima are equal to zero yields that it suffices to show that κh[u] has at most one sign change in[0, 1).

We assume by contradiction that there exist 0 < x1 < x0 < x2 ≤ 1 such that κh[u] > 0 in(x1, x0), κh[u] < 0 in (x0, x2) and ϕ(x1) = ϕ(x2). We construct a new function with lower Willmoreenergy. This new function equals the original one on [0, x1]. Then we take the arc of the circle withcentre (ϕ(x1), 0) and radius r(x1), starting at (x1, u(x1)) and ending where this arc intersects thestraight line which connects (ϕ(x1), 0) = (ϕ(x2), 0) and (x2, u(x2)). Finally, we attach the suitably

52

rescaled original function u|[x2,1]. More precisely, with the scaling factor = r(x1)/r(x2) the newfunction is

v(x) :=

u(x) if 0 ≤ x ≤ x1,√

r(x1)2 − (x − ϕ(x1))2 if x1 ≤ x ≤ x2 + u′(x2)u(x2)(1 − ),

u(

1

(x − (1 − )ϕ(x2)

))if x2 + u′(x2)u(x2)(1 − ) ≤ x ≤ + (1 − )ϕ(x2),

(6.4)

and extended by symmetry to [−ℓ(), ℓ()], setting ℓ() := + (1 − )ϕ(x2). See Figure 13. Thefunction v satisfies v ∈ C1,1([−ℓ(), ℓ()], (0,∞)), v(ℓ()) = α and v′(ℓ()) = −β.

a)

κh < 0

κh > 0

κh < 0

α

radius to (ϕ(x1), 0)

radius to (ϕ(x2), 0)

x1x0 x2 b)

κh > 0 κh < 0

α

x1 x0 x2

Figure 13: Proof of Theorem 6.4.

Note that ≤ 1. Indeed, one may write

r(x2) = r(x1) +

x2∫

x1

u′(t)√1 + u′(t)2

ϕ′(t) dt. (6.5)

By our assumption on ϕ and since u′(x) ≤ 0 in [0, 1], the integrand in (6.5) is negative in [x1, x0]and positive in [x0, x2]. Moreover, since p 7→ p√

1+p2is strictly increasing and, by Lemma 6.3, u′ is

strictly decreasing in [0, 1], also u′√1+u′2

is monotonically decreasing. Hence, splitting the integral

in (6.5) and using that ϕ(x1) = ϕ(x2) we get

r(x2) > r(x1) +u′(x0)√

1 + u′(x0)2

x0∫

x1

ϕ′(t) dt +u′(x0)√

1 + u′(x0)2

x2∫

x0

ϕ′(t) dt = r(x1).

By scaling, the function w(x) = 1ℓ() v

(ℓ()x

)defined in [−1, 1], satisfies

w′(1) = −β, w(1) =α

ℓ()and Wh(w) < Wh(u). (6.6)

53

On the other hand, < 1, ϕ(x2) < 0 give > l(), and so w(1) > α = u(1). By monotonicity ofthe hyperbolic Willmore energy (see Proposition 3.12) we conclude that Wh(w) > Wh(u). Thiscontradicts (6.6).

6.2 The case αβ < 1 and β ≥ 0


Proposition 6.5 (Upper bound for the energy). We have

Mα,β ≤ min2(1 − αβ)√

1 + β2

1

α, 8 tanh

(√1 + β2

α+ arsinh(β)

)− 8

β√1 + β2

.

Proof. For β > 0, the first estimate follows with the same construction as in Proposition 6.1

considering the function w ∈ Nα,β given by (6.1); w(x) = α − 1β +

√1 + 1

β2 − x2. Computing its

Willmore energy we find

Wh(w) =αβ − 1√1 + β2

1∫

−1

1√

1 + 1β2 − x2

− 1

α − 1β +

√1 + 1

β2 − x2

dx

≤ 1 − αβ√1 + β2

1∫

−1

1

α − 1β +

√1 + 1

β2 − x2dx ≤ 1 − αβ√

1 + β2

2

α.

If β = 0, the function w(x) ≡ α directly gives Mα,0 ≤ 2α . Let fα the function defined in (5.1).

Notice that it is well defined since we assume that β ≥ 0 and αβ < 1. From the calculations onp. 45 we see that

Mα,β ≤ Wh(fα) ≤ 8 tanh

(√1 + β2

αx1

)− 8

β√1 + β2

,

where x1 = 1 + α√1+β2

arsinh(β).

In the special case β = 0 we may now characterise the asymptotic behaviour of Mα,0 for α → ∞.This is completely different from the case β 6= 0, cf. Proposition 6.2.

Proposition 6.6. We assume that α > 0 and β = 0. Then, one has

2α

(α + 1)√

1 + α2≤ Mα,0 ≤ 8 tanh

(1

α

).

Proof. Let uα ∈ Nα,0 be a minimiser for Wh in Nα,0 as constructed in Theorem 3.18 . It satisfiesuα(x) ≤ α + 1 and |u′

α(x)| ≤ 1/α, which yields

Mα,0 = Wh(uα) ≥ 2

1∫

0

1

uα(x)√

1 + u′α(x)2

dx ≥ 2α

(α + 1)√

1 + α2,

which is the estimate from below. The estimate from above was just proved in Proposition 6.5.

54


Theorem 6.7. We assume that β ≥ 0 and that αβ < 1. Let u ∈ Nα,β be a minimiser for Mα,β.Then, κh[u] > 0 in (−1, 1).

Proof. The proof is along the lines of Theorem 6.4. We recall the main points and emphasise onwhat is different. We associate to u the auxiliary function ϕ(x) := x+u(x)u′(x), x ∈ [−1, 1]. From(3.4) we know that ϕ(0) = 0 and ϕ(x) > 0 in (0, 1]. Since ϕ cannot be constant in an interval,ϕ(x) increases in a right neighbourhood of 0 and hence κh[u](x) > 0 in a right neighbourhood of0.

We now prove that ϕ′ ≥ 0 in [0, 1]. If ϕ′ < 0 in some interval then there exist points 0 <x0 < x′ < 1 such that ϕ′ ≥ 0 (i.e. κh[u] ≥ 0) in [0, x0] and ϕ′ < 0 (i.e. κh[u] < 0) in (x0, x

′). Inparticular, u′′ < 0 in [x0, x

′]. Let x∗ ∈ [0, x0) be such that u′′(x) < 0 in (x∗, x′]. Then, there existx1 ∈ [x∗, x0) and x2 ∈ (x0, x

′] such that ϕ(x1) = ϕ(x2).Then, with the same notation as in the proof of Theorem 6.4, we consider the function v ∈

C1,1([−ℓ(), ℓ()], (0,∞)) defined as in (6.4). Then, w(x) := 1ℓ()v(ℓ()x), x ∈ [−1, 1], satisfies

w′(1) = −β, w(1) =α

ℓ()and Wh(w) < Wh(u). (6.7)

Since u′′ < 0 in [x1, x2] and proceeding as in (6.5) one sees that < 1. Here, in contrast with theproof of Theorem 6.4, ϕ > 0 in (0, 1) and so < ℓ(). Hence, w(1) < u(1) and Wh(w) > Wh(u) bythe strict monotonicity of the energy (Proposition 3.19), which contradicts the inequality in (6.7).Then, ϕ being increasing in [0, 1] implies that κh[u] ≥ 0 in [−1, 1]. The strong minimum principlefor (2.5) considered as a second order equation for κh[u] and applied to a possible minimum 0yields that κh[u] > 0 in (−1, 1).

6.3 The case β < 0 and α ≥ αβ


Proposition 6.8 (Upper bound of the energy). We have

Mα,β ≤ (1 + arsinh(−β)(α − αβ))(−8β)√1 + β2

.

Proof. If α = αβ , the minimiser is uc(x) = cosh(bx)/b, b = arsinh(−β), which has vanishing meancurvature in [−1, 1] and hyperbolic Willmore energy

Wh(uc) = − 8β√1 + β2

.

If instead α > αβ , we consider the function u := uc + δα where δα := α − αβ > 0. Notice thatu ∈ Nα,β . Since uc ≥ 1

b > 0 and u′′c > 0 we have

Wh(uc + δα) =

1∫

−1

(u′′

c

(1 + u′2c )

3

2

+1

(uc + δα)√

1 + u′2c

)2

(uc + δα)√

1 + u′2c dx

≤ Wh(uc) +

1∫

−1

(u′′

c

(1 + u′2c )

3

2

+1

uc

√1 + u′2

c

)2

δα

√1 + u′2

c dx

≤ Wh(uc) + bδα

1∫

−1

(u′′

c

(1 + u′2c )

3

2

+1

uc

√1 + u′2

c

)2

uc

√1 + u′2

c dx

55

which proves the proposition.

The previous result together with Proposition 6.2 shows that also for β < 0, Mα,β grows linearlyfor α → ∞.


Here we prefer a slightly less general formulation of the curvature statement and refer only tosolutions as we have constructed. The reason is that we have to restrict the set of admissiblefunctions in the case −β < α.

Theorem 6.9. We assume that β < 0 and that α ≥ αβ. Let u ∈ Nα,β be an energy minimisingsolution of (1.4) as constructed in the proofs of Theorems 4.17 and 4.24. Then, either κh[u] > 0in [0, 1), or there exists a point a ∈ (0, 1) such that κh[u] > 0 in [0, a) and κh[u] < 0 in (a, 1).

Proof. We have u′ > 0 in (0, 1]. We associate to u the function ϕ(x) := x + u(x)u′(x), x ∈ [−1, 1],which satisfies ϕ(0) = 0 and ϕ(x) > 0 in (0, 1]. Hence, ϕ(x) increases in a right neighbourhood of0 and so, κh[u](x) > 0 in a right neighbourhood of 0. In view of the strong maximum /minimumprinciple for (2.5) as a second order equation for κh[u] applied to maximum /minimum equal to 0,we only need to exclude that there is a sign change from κh[u] < 0 to κh[u] > 0.

We assume by contradiction that there exist 0 < x < x0 < x′ such that κh[u] < 0 (i.e. ϕ′ < 0)in (x, x0) and κh[u] > 0 (i.e. ϕ′ > 0) in (x0, x

′). In particular, u′′ < 0 in [x, x0] and so, alsoon a slightly larger interval [x, x′′) ⊃ [x, x0]. Then, we find x1 ∈ (x, x0) and x2 ∈ (x0, x

′′) withϕ(x1) = ϕ(x2).

We consider the function v ∈ C1,1([−ℓ(), ℓ()], (0,∞)) as defined in (6.4). Using the samenotation as in the proof of Theorem 6.4, one should notice that in this case > 1. Indeed, onestarts from (6.5). By our assumption on ϕ and since u′(x) > 0 in (0, 1], the integrand in (6.5) isstrictly negative in (x1, x0) and positive in (x0, x2). Moreover since u′ is decreasing in (x1, x2),also u′√

1+u′2is monotonically decreasing. Hence, splitting the integral in (6.5) and using that

ϕ(x1) = ϕ(x2) we get

r(x2) < r(x1) +u′(x0)√

1 + u′(x0)2

x0∫

x1

ϕ′(t) dt +u′(x0)√

1 + u′(x0)2

x2∫

x0

ϕ′(t) dt = r(x1).

Notice that even though > 1, x2 + u(x2)u′(x2)(1 − ) ≥ x1.

By scaling, the function w(x) = 1ℓ() v

(ℓ()x

)defined in [−1, 1] satisfies

w′(1) = −β, w(1) =α

ℓ()and Wh(w) < Wh(u). (6.8)

On the other hand, since ϕ > 0 on (0, 1] we have > l(), w(1) > u(1) and hence Wh(w) ≥Wh(u) by monotonicity of the hyperbolic Willmore energy (see Propositions 4.18 and 4.25), whichcontradicts the inequality in (6.8).

6.4 The case β < 0 and α < αβ


Proposition 6.10 (Upper bound of the energy). The following estimate holds

Mα,β ≤ − 8β√1 + β2

+ 8 tanh

(arsinh(−β)

α(αβ − α)

).

56

Proof. For β < 0 and α < αβ the function fα defined in (5.1) is well defined. As usual we denoteb = arsinh(−β). The claim follows then from

Mα,β ≤ Wh(fα) = − 8β√1 + β2

− 8 tanh

(bαβ

α(x0 − 1 +

α

αβ)

)

≤ − 8β√1 + β2

+ 8 tanh

(bαβ

α(1 − α

αβ)

).


Theorem 6.11. Let u ∈ Nα,β be a minimiser for Mα,β. Then, κh[u] > 0 in (−1, 1).

Proof. We know that there exists x0 ∈ [0, 1) such that u′(x0) = 0, u′ > 0 in (x0, 1] and u′ < 0in (0, x0). Then, u|[−x0,x0] is the rescaled minimiser of Mu(x0)/x0,0 and hence, by Theorem 6.7,κh[u] > 0 in (−x0, x0).

It remains to study the sign of the curvature in [x0, 1]. Again, we consider ϕ(x) := x+u(x)u′(x),x ∈ [x0, 1]. Since u′ > 0 in (x0, 1], ϕ(x0) = x0 and ϕ(x) > x0 in (x0, 1], ϕ(x) increases in a rightneighbourhood of x0 and hence, κh[u](x) > 0 also in a right neighbourhood of x0.

In view of the strong minimum principle of (2.5) considered as a second order equation forκh[u] and applied to a possible minimum 0, is suffices to show that κh[u] ≥ 0 everywhere. Weassume by contradiction that there exist x0 < x′ < x′′ < 1 such that κh[u] > 0 (i.e. ϕ′ > 0) in[0, x′) and κh[u] < 0 (i.e. ϕ′ < 0) in (x′, x′′). In particular, u′′ < 0 in [x′, x′′) and so, also on aslightly larger interval (x∗, x′′) ⊃ [x′, x′′). Finally, we may find x1 ∈ (x∗, x′) and x2 ∈ (x′, x′′) withϕ(x1) = ϕ(x2).

Then, with the same notation as in the proof of Theorem 6.4, we consider the function v ∈C1,1([−ℓ(), ℓ()], (0,∞)) defined as in (6.4). Notice that in this case < 1. Indeed, by ourassumption on ϕ and since u′(x) > 0 in (x0, 1], the integrand in (6.5) is strictly positive in[x1, x

′) and negative in (x′, x2]. Moreover since u′ is strictly decreasing in [x1, x2], also u′√1+u′2

is monotonically decreasing. Splitting the integral in (6.5) and using that ϕ(x1) = ϕ(x2) we get

r(x2) > r(x1) +u′(x′)√

1 + u′(x′)2

x′∫

x1

ϕ′(t) dt +u′(x′)√

1 + u′(x′)2

x2∫

x′

ϕ′(t) dt = r(x1).

Then, w(x) := 1ℓ()v(ℓ()x), x ∈ [−1, 1], satisfies

w′(1) = −β, w(1) =α

ℓ()and Wh(w) < Wh(u). (6.9)

Since ϕ > 0 on (0, 1] we have < ℓ() and w(1) < u(1). Therefore, Wh(w) > Wh(u) bymonotonicity of the energy (see Propositions 4.40 and 4.49), which contradicts the inequalityin (6.9).

7 Numerical studies and algorithms

We will try to approximate a solution u(x) of the Willmore equation (1.4) by approximating thestationary limit u(x) = limt→∞ U(x, t) of the solution U(x, t) of the Dirichlet problem of the

57

Willmore flow equation

V = ΓH + 2H3 − 2HK ∀x ∈ (−1, 1) , t > 0,

U(−1, t) = U(1, t) = α, Ux(−1, t) = −Ux(1, t) = β ∀t > 0,

U(x, 0) = u0(x) ∀x ∈ [−1, 1],

(7.10)

for a family (Γ(t))t∈[0,∞) of axially symmetric surfaces parametrised by

Γ(t) := (x, U(x, t) cosϕ, U(x, t) sinϕ) : x ∈ [−1, 1], ϕ ∈ [0, 2π] .

Here H, K denote the quantities related to the surface Γ(t) generated by the function U(x, t) andV the normal velocity of Γ(t) given by

V =Ut

(1 + U2x)1/2

.

In order to derive the variational formulation of (7.10) we exploit the fact that the right handside of (7.10) is linked with the derivative of the Willmore functional

W(Γ) =

∫

ΓH2 dA.

In fact, writing W(U) instead of W(Γ), we can show that

W(U) = 2πW(U) + 2πβ√

1 + β2(7.11)

where

W(U) :=1

4

∫ +1

−1

UU2

xx

(1 + U2x)5/2

+1

U(1 + U2x)1/2

dx.

Thus, for the derivative in direction φ ∈ H20 (−1, 1), we obtain

〈 W ′(U), φ 〉 := ddε W(U + εφ)

∣∣ε=0

=1

4

∫ +1

−1

2

UUxxφxx

(1 + U2x)5/2

+U2

xxφ

(1 + U2x)5/2

− 5UUxU2

xxφx

(1 + U2x)7/2

− φ

U2(1 + U2x)1/2

− Uxφx

U(1 + U2x)3/2

dx.

(7.12)

Under the smoothness assumption U ∈ H4(−1, 1) one can prove (see §2.3) that

〈 W ′(U), φ 〉 = −∫ +1

−1Uφ

(ΓH + 2H3 − 2HK

)dx ∀φ ∈ H2

0 (−1, 1). (7.13)

Multiplying (7.10) with the test function Uφ and integrating over [−1, 1] yields the followingvariational formulation :

For t ≥ 0 find U(·, t) ∈ X such that U(., 0) = u0 and∫ +1

−1

U(·, t)Ut(·, t)φ(1 + Ux(·, t)2)1/2

dx + 〈 W ′(U(·, t)), φ 〉 = 0 ∀φ ∈ H20 (−1, 1), t > 0,

(7.14)

58

where 〈 W ′(U(·, t)), φ 〉 is defined by (7.12) and

X := v ∈ H2(−1, 1) : v(−1) = v(1) = α, v′(−1) = −v′(1) = β.

For the numerical solution of (7.14), we use the finite element method to get a finite dimensionalnonlinear system of ODEs. To this end, we decompose the space interval I = [−1, 1] into elementsKi = [xi−1, xi], i = 1, . . . , N , and define the finite element space Xh ⊂ X as

Xh := v ∈ X : v∣∣Ki

∈ P3 ∀ i = 1, . . . , N, (7.15)

where P3 denotes the space of polynomials with degree less or equal to 3. Note that Xh ⊂ C1(I, R),i.e. we use C1-elements of third order. The degrees of freedom are the values of the function andof the first derivative at the nodes xi where the values at the boundary points x0 = −1 and xN = 1are prescribed by the values α and β due to Xh ⊂ X. Thus, the semi-discrete solution Uh(·, t) ∈ Xh

of problem (7.14) can be represented as

Uh(x, t) =2N+2∑

j=1

cj(t)ϕj(x), (7.16)

where the basis functions ϕj ∈ C1(I, R) are defined as follows. For each element Ki, it holdsϕj

∣∣Ki

∈ P3 for all j. The first set of basis functions ϕ1+i, i = 0, . . . , N , is responsible for the pointvalues of the discrete function at the nodes xi, i.e., it holds

ϕ1+i(xk) = δi,k,∂

∂xϕ1+i(xk) = 0 ∀ k = 0, . . . , N, i = 0, . . . N.

The second set ϕN+2+i is responsible for the values of the x-derivatives at the nodes xi, i.e., itholds

ϕN+2+i(xk) = 0,∂

∂xϕN+2+i(xk) = δi,k ∀ k = 0, . . . , N, i = 0, . . . N.

These conditions for the definition of the basis functions ϕj imply the following meaning of thecoefficients cj(t) in (7.16)

c1+i(t) = Uh(xi, t), cN+2+i(t) =∂

∂xUh(xi, t) ∀ i = 0, . . . , N,

where c1(t) = cN+1(t) = α and cN+2(t) = −c2N+2(t) = β for all t > 0 due to the Dirichletboundary conditions of Uh(x, t). Note that the support of the basis functions ϕ1+i and ϕN+2+i islocal; it consists of the (at most two) elements that contain the node xi. The initial condition isdiscretised as Uh(x, 0) = u0,h(x) =

∑2N+2j=1 c0,jϕj(x), where u0,h ∈ Xh is a suitable interpolant of u0

in Xh, which can be defined, for instance, by the choice c0,1+i := u0(xi) and c0,N+2+i := ∂∂xu0(xi)

for i = 0, . . . , N . Therefore, we have the initial conditions cj(0) = c0,j for the unknown coefficientfunctions cj(t). For the discrete problem, we need the test space

Xh,0 := v ∈ H20 (−1, 1) : v

∣∣Ki

∈ P3 ∀ i = 1, . . . , N.

Then, the semi-discrete variational problem reads :

For t ≥ 0 find Uh(·, t) ∈ Xh such that Uh(., 0) = u0,h and∫ +1

−1

Uh(·, t)Uht(·, t)φh

(1 + Uhx(·, t)2)1/2dx + 〈 W ′(Uh(·, t)), φh 〉 = 0 ∀φh ∈ Xh,0, t > 0.

(7.17)

59

Using the ansatz (7.16) for Uh(·, t) and taking the test functions φh = ϕi ∈ Xh,0 for i ∈ Jh :=1, . . . , 2N +2\1, N +1, N +2, 2N +2, we see that (7.17) is equivalent to a nonlinear (2N −2)-dimensional system of ODEs for the coefficient functions cj(t), j ∈ Jh .

In the time discretisation, we calculate for a discrete time level tn an approximation Un ∈ Xh

of Uh(·, tn). Starting with t0 = 0 and U0 := u0,h, we assume that Un is known. We choose a timestep kn > 0 and compute Un+1 at the time level tn+1 := tn + kn from (7.17) by approximating thetime derivative Uht at t = tn by the first order backward difference formula

Uht(tn) ≈ Un+1 − Un

kn.

In order to get a linear system of equations for Un+1 we replace in (7.17) several nonlinear termsof Uh(·, t) by the known function Un ∈ Xh, i.e., we compute Un+1 ∈ Xh from

∫ +1

−1

Un

(1 + (Unx )2)1/2

(Un+1 − Un

)φhdx + kn〈 Wn

′(Un+1), φh 〉 = 0 ∀φh ∈ Sh, (7.18)

where Wn′(Un+1) is the following linear approximation of W ′(Un+1) :

〈 Wn′(Un+1), φh 〉 :=

1

4

∫ +1

−1

2

UnUn+1xx φhxx

(1 + (Unx )2)5/2

+(Un

xx)2φh

(1 + (Unx )2)5/2

− 5Un(Un

xx)2Un+1x φhx

(1 + (Unx )2)7/2

− φh

(Un)2(1 + (Unx )2)1/2

− Un+1x φhx

Un(1 + (Unx )2)3/2

dx.

Note that the places, where we have taken Un and where Un+1, have been chosen heuristically.Other choices are possible and will be studied in future.

Numerical experiments have shown that the choice of a constant time step kn = t for alln = 0, 1, . . . , does not lead to satisfying results. If the time step is too large the sequence Uncan be divergent and if it is too small one needs a very large computing time to reach a stationarylimit. Therefore, we have developed an adaptive time step control which is presented in Figure 14.In the step (b2) we discard the computed solution Un+1 if its energy has increased compared withUn or if the relative change of the energy was too large. We divide the time step size by two andcompute Un+1 again. In all other cases we accept the solution Un+1. Moreover, we double the sizefor the next time step if the relative change of the energy was at least in two previous time stepstoo small and the doubled size is not larger than a prescribed value kmax.

In our numerical experiments, we have chosen the control parameters ωmax = 0.1, ωmin = 0.01,kmax = 0.01 (see Figure 14) and the initial time step size k0 = h4 where h := max1≤i≤N |xi −xi−1|denotes the mesh-size. We accept Un+1 to be the stationary limit of our time marching algorithmif

k−1n |W(Un+1) −W(Un)| < εW and k−1

n ‖Un+1 − Un‖∞ < εU , (7.19)

where the norm ‖ · ‖∞ is defined as

‖Uh‖∞ := max1≤j≤2N+2

|cj | for Uh =2N+2∑

j=1

cjϕj ∈ Xh,

and εW , εU are some prescribed tolerances which have been chosen as εW = εU = 10−4. We stopthe time iteration if the criterion (7.19) is satisfied or if, in case (b2) of the adaptive time stepcontrol (Figure 14), for the halved time step size it holds kn/2 <

√eps, where eps is the relative

floating-point accuracy, or if a maximum number nmax of iteration steps is reached (we have usednmax = 360000).

60

Computation of (tn+1, kn+1, Un+1) from (tn, kn, Un) :

(a) initialize: W0 := W(Un) (formula (7.11)); T accept := 0 ; N too small := 0 ;

(b) while T accept = 0 do

(1) compute: Un+1 by solving (7.18) ; W := W(Un+1) ;

(2) if W > W0 or |W −W0| > ωmax|W0| then

kn := kn/2 ;

(3) else if |W −W0| < ωmin|W0| and 2kn < kmax then

T accept := 1 ;

if N too small ≥ 2 then

kn := 2kn ; N too small := 0 ;

else

N too small := N too small + 1 ;endif

(4) else

T accept := 1 ;endif

enddo

(c) kn+1 := kn ; tn+1 := tn + kn ;

Figure 14: Time marching algorithm with control parameters ωmax, ωmin and kmax.

In the following, we describe our numerical results for three different settings of the boundarydata α and β. In the first case, we combine the value α = 0.5 with the three negative valuesβ ∈ −1,−5,−10, see Figure 15. In the second case, we consider α = 0.5 and the positive valuesβ ∈ 1, 5, 10, see Figure 16. Finally, in the third case, we study the situations where β = 0 andα ∈ 0.5, 0.1, 0.01, see Figure 17. The finite element mesh depends on the data α and β and islocally adapted near the boundary in the following way. The interval I = [−1, 1] is decomposedinto the three subintervals Ω1 = [−1,−1 + δ], Ω2 = [−1 + δ, 1 − δ] and Ω3 = [1 − δ, 1] where

δ := min2

3, ω0 min|α|, 1

|β| + 10−6

with ω0 = 2 except for the case (α, β) = (0.01, 0) where ω0 = 8. Then the mesh is created bysubdividing each subinterval Ωk into N0 = 40 equidistant elements. In each picture of Figures 15 -17, the mesh is shown on the x-axis, the initial solution U0 at t0 = 0 is presented by the dottedline and the final solution Un at the end tn of the time iteration is given by the solid line. Theparameters at the headline of the picture have the following meaning. n denotes the number ofthe last time step, dt the last time step size kn, NEL the number of the finite elements and W thevalue of the Willmore functional W(Un).

For nearly all cases, our numerical algorithm produced a discrete solution satisfying the approx-imate “stationarity” criterion (7.19) with the tolerances εW = εU = 10−4. The first exceptionalcase was (α, β) = (0.5,−10) where the algorithm at a time tn could not find a next function Un+1

at a time tn + kn with kn >√

eps such that W(Un+1) < W(Un). The other critical case was(α, β) = (0.01, 0) where for all n a new Un+1 with a smaller value of the Willmore functional

61

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

dt = 5.06e−03, NEL = 120, n = 116, tn = 3.209e−01, W = 5.535e+00

β = −1

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

dt = 5.24e−03, NEL = 120, n = 262, tn = 7.671e−01, W = 8.523e+00

β = −5

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

dt = 2.00e−08, NEL = 120, n = 2864, tn = 4.683e−01, W = 9.258e+00

β = −10 : k−1n ‖Un+1 − Un‖∞ ≈ 1.67e − 02, stopped by kn/2 <

√eps

Figure 15: α = 0.5 and β < 0

62

−1 −0.5 0 0.5 10

0.5

1

1.5

dt = 5.06e−03, NEL = 120, n = 261, tn = 1.092e+00, W = 9.430e+00

β = 1

−1 −0.5 0 0.5 10

0.5

1

1.5

dt = 5.24e−03, NEL = 120, n = 642, tn = 3.083e+00, W = 1.270e+01

β = 5

−1 −0.5 0 0.5 10

0.5

1

1.5

dt = 5.24e−03, NEL = 120, n = 863, tn = 4.179e+00, W = 1.320e+01

β = 10

Figure 16: α = 0.5 and β > 0

63

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dt = 5.06e−03, NEL = 120, n = 185, tn = 7.029e−01, W = 5.861e+00

α = 0.5

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dt = 5.24e−03, NEL = 120, n = 372, tn = 1.604e+00, W = 1.113e+01

α = 0.1

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dt = 5.24e−07, NEL = 120, n = 360000, tn = 1.447e−01, W = 1.244e+01

α = 0.01 : stopped by n = nmax, ‖Un+1 − Un‖∞ ≈ 2.75e − 04

Figure 17: β = 0

64

could be found but where the maximum number of nmax = 360000 time steps was reached with-out satisfying the criterion (7.19). On the other hand, the change of the graph of the discretesolution as well as the time step size over the last 99% of all time steps was very small. Here,further research is necessary to figure out the reason for this behaviour. Could it be that the firstorder time discretisation is not accurate enough or that the semi-implicit backward Euler exhibitssome instabilities which lead to very small time steps? Another reason could be that the “exact”continuous Willmore flow is really creeping very slowly to the stationary limit. Let us finally notethat, for such critical cases, a suitable choice of the initial function U0 = u0,h is important for theconvergence of the numerical solution to the stationary limit. Here a good analytical feeling isvery helpful. We construct u0,h ∈ Xh as the interpolant of a suitable function u0 ∈ X as describedabove. In the case β ≤ 0, we use u0 = fα with fα defined in (5.1) which is a catenoid at theboundary fitted with an arc of a circle centered at the origin. For β > 0, we choose u0 = w withw defined in (6.1) which is an arc of a circle centered at (0, α − 1/β).

Acknowledgment. The third author is grateful to G. Huisken (Max-Planck-Institute Golm) forsuggesting to study boundary value problems for the Willmore equation, to E. Kuwert for pointingout the importance of Reference [LS2] concerning Willmore surfaces of revolution. We thank M.Bergner, K. Deckelnick and R. Schatzle for interesting discussions.

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Anna Dall’Acqua, Fakultat fur Mathematik, Otto-von-Guericke-Universitat, Postfach 4120, D-39016 Magdeburg, GermanyE-mail: [email protected]

Steffen Frohlich, Institut fur Mathematik, Fachbereich Mathematik und Informatik, Freie Univer-sitat Berlin, Arnimallee 3, D-14195 Berlin, GermanyE-mail: [email protected]

Hans-Christoph Grunau, Fakultat fur Mathematik, Otto-von-Guericke-Universitat, Postfach 4120,D-39016 Magdeburg, GermanyE-mail: [email protected]

Friedhelm Schieweck, Fakultat fur Mathematik, Otto-von-Guericke-Universitat, Postfach 4120, D-39016 Magdeburg, GermanyE-mail: [email protected]

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symmetric willmore surfaces of revolution satisfying

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